Dirichlet series
| L(s) = 1 | − 8·2-s + 21·3-s + 15·4-s − 28·5-s − 168·6-s − 59·7-s + 33·8-s + 252·9-s + 224·10-s − 131·11-s + 315·12-s − 123·13-s + 472·14-s − 588·15-s − 81·16-s − 235·17-s − 2.01e3·18-s − 80·19-s − 420·20-s − 1.23e3·21-s + 1.04e3·22-s − 274·23-s + 693·24-s + 51·25-s + 984·26-s + 2.26e3·27-s − 885·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4.04·3-s + 15/8·4-s − 2.50·5-s − 11.4·6-s − 3.18·7-s + 1.45·8-s + 28/3·9-s + 7.08·10-s − 3.59·11-s + 7.57·12-s − 2.62·13-s + 9.01·14-s − 10.1·15-s − 1.26·16-s − 3.35·17-s − 26.3·18-s − 0.965·19-s − 4.69·20-s − 12.8·21-s + 10.1·22-s − 2.48·23-s + 5.89·24-s + 0.407·25-s + 7.42·26-s + 16.1·27-s − 5.97·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 59^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 59^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 59^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.35480\times 10^{7}\) |
| Root analytic conductor: | \(3.23161\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 3^{7} \cdot 59^{7} ,\ ( \ : [3/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T )^{7} \) |
| 59 | \( ( 1 + p T )^{7} \) | |
| good | 2 | \( 1 + p^{3} T + 49 T^{2} + 239 T^{3} + 497 p T^{4} + 3677 T^{5} + 3001 p^{2} T^{6} + 8897 p^{2} T^{7} + 3001 p^{5} T^{8} + 3677 p^{6} T^{9} + 497 p^{10} T^{10} + 239 p^{12} T^{11} + 49 p^{15} T^{12} + p^{21} T^{13} + p^{21} T^{14} \) |
| 5 | \( 1 + 28 T + 733 T^{2} + 13236 T^{3} + 45836 p T^{4} + 658188 p T^{5} + 43307182 T^{6} + 502313288 T^{7} + 43307182 p^{3} T^{8} + 658188 p^{7} T^{9} + 45836 p^{10} T^{10} + 13236 p^{12} T^{11} + 733 p^{15} T^{12} + 28 p^{18} T^{13} + p^{21} T^{14} \) | |
| 7 | \( 1 + 59 T + 2484 T^{2} + 233 p^{3} T^{3} + 2291665 T^{4} + 56354724 T^{5} + 1239033202 T^{6} + 23961363956 T^{7} + 1239033202 p^{3} T^{8} + 56354724 p^{6} T^{9} + 2291665 p^{9} T^{10} + 233 p^{15} T^{11} + 2484 p^{15} T^{12} + 59 p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 + 131 T + 11811 T^{2} + 752384 T^{3} + 39426912 T^{4} + 1775454192 T^{5} + 71672161798 T^{6} + 246690517014 p T^{7} + 71672161798 p^{3} T^{8} + 1775454192 p^{6} T^{9} + 39426912 p^{9} T^{10} + 752384 p^{12} T^{11} + 11811 p^{15} T^{12} + 131 p^{18} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 + 123 T + 17801 T^{2} + 1447080 T^{3} + 122306176 T^{4} + 7406135196 T^{5} + 453070569650 T^{6} + 21198782985110 T^{7} + 453070569650 p^{3} T^{8} + 7406135196 p^{6} T^{9} + 122306176 p^{9} T^{10} + 1447080 p^{12} T^{11} + 17801 p^{15} T^{12} + 123 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 + 235 T + 25532 T^{2} + 1021245 T^{3} - 25012825 T^{4} - 1157257352 T^{5} + 674098575784 T^{6} + 78971566559480 T^{7} + 674098575784 p^{3} T^{8} - 1157257352 p^{6} T^{9} - 25012825 p^{9} T^{10} + 1021245 p^{12} T^{11} + 25532 p^{15} T^{12} + 235 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 + 80 T + 13628 T^{2} + 874406 T^{3} + 195792705 T^{4} + 14602997348 T^{5} + 1654296222858 T^{6} + 86465262355916 T^{7} + 1654296222858 p^{3} T^{8} + 14602997348 p^{6} T^{9} + 195792705 p^{9} T^{10} + 874406 p^{12} T^{11} + 13628 p^{15} T^{12} + 80 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 + 274 T + 51194 T^{2} + 8423538 T^{3} + 1378971477 T^{4} + 192030161988 T^{5} + 23789786316232 T^{6} + 2667592051274736 T^{7} + 23789786316232 p^{3} T^{8} + 192030161988 p^{6} T^{9} + 1378971477 p^{9} T^{10} + 8423538 p^{12} T^{11} + 51194 p^{15} T^{12} + 274 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 + 14 p T + 150112 T^{2} + 33348906 T^{3} + 6381579931 T^{4} + 918039787080 T^{5} + 121903392565336 T^{6} + 16989021113948792 T^{7} + 121903392565336 p^{3} T^{8} + 918039787080 p^{6} T^{9} + 6381579931 p^{9} T^{10} + 33348906 p^{12} T^{11} + 150112 p^{15} T^{12} + 14 p^{19} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 + 346 T + 174888 T^{2} + 43226778 T^{3} + 13301202123 T^{4} + 84602853962 p T^{5} + 611160453023164 T^{6} + 97904675540052172 T^{7} + 611160453023164 p^{3} T^{8} + 84602853962 p^{7} T^{9} + 13301202123 p^{9} T^{10} + 43226778 p^{12} T^{11} + 174888 p^{15} T^{12} + 346 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 + 157 T + 72646 T^{2} + 22184595 T^{3} + 5166449981 T^{4} + 1797892907318 T^{5} + 401789055250908 T^{6} + 87178153061421944 T^{7} + 401789055250908 p^{3} T^{8} + 1797892907318 p^{6} T^{9} + 5166449981 p^{9} T^{10} + 22184595 p^{12} T^{11} + 72646 p^{15} T^{12} + 157 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 + 825 T + 472232 T^{2} + 210376935 T^{3} + 85993284187 T^{4} + 30147869740300 T^{5} + 9301444129405536 T^{6} + 2540902693719103296 T^{7} + 9301444129405536 p^{3} T^{8} + 30147869740300 p^{6} T^{9} + 85993284187 p^{9} T^{10} + 210376935 p^{12} T^{11} + 472232 p^{15} T^{12} + 825 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 + 815 T + 633215 T^{2} + 294771820 T^{3} + 3000832248 p T^{4} + 42629738011060 T^{5} + 13987023495260402 T^{6} + 3864432785061708458 T^{7} + 13987023495260402 p^{3} T^{8} + 42629738011060 p^{6} T^{9} + 3000832248 p^{10} T^{10} + 294771820 p^{12} T^{11} + 633215 p^{15} T^{12} + 815 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 1196 T + 1219398 T^{2} + 824403468 T^{3} + 483639309677 T^{4} + 225526943068976 T^{5} + 92690939829185540 T^{6} + 31733592361461877744 T^{7} + 92690939829185540 p^{3} T^{8} + 225526943068976 p^{6} T^{9} + 483639309677 p^{9} T^{10} + 824403468 p^{12} T^{11} + 1219398 p^{15} T^{12} + 1196 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 + 900 T + 799937 T^{2} + 518231056 T^{3} + 298189356028 T^{4} + 148848017614048 T^{5} + 68783675234747738 T^{6} + 27302985098307656008 T^{7} + 68783675234747738 p^{3} T^{8} + 148848017614048 p^{6} T^{9} + 298189356028 p^{9} T^{10} + 518231056 p^{12} T^{11} + 799937 p^{15} T^{12} + 900 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 - 420 T + 1115532 T^{2} - 7700114 p T^{3} + 579877809621 T^{4} - 233378093269662 T^{5} + 189357032417163714 T^{6} - 67368433292137469184 T^{7} + 189357032417163714 p^{3} T^{8} - 233378093269662 p^{6} T^{9} + 579877809621 p^{9} T^{10} - 7700114 p^{13} T^{11} + 1115532 p^{15} T^{12} - 420 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 - 1316 T + 2252319 T^{2} - 2017101760 T^{3} + 2040365018688 T^{4} - 1384565546906776 T^{5} + 1017033483907028802 T^{6} - \)\(53\!\cdots\!72\)\( T^{7} + 1017033483907028802 p^{3} T^{8} - 1384565546906776 p^{6} T^{9} + 2040365018688 p^{9} T^{10} - 2017101760 p^{12} T^{11} + 2252319 p^{15} T^{12} - 1316 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 + 173 T + 1800755 T^{2} + 302758388 T^{3} + 1558723799676 T^{4} + 233952228243424 T^{5} + 837923811989445386 T^{6} + \)\(10\!\cdots\!10\)\( T^{7} + 837923811989445386 p^{3} T^{8} + 233952228243424 p^{6} T^{9} + 1558723799676 p^{9} T^{10} + 302758388 p^{12} T^{11} + 1800755 p^{15} T^{12} + 173 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 + 418 T + 1496388 T^{2} + 491184668 T^{3} + 1202670029317 T^{4} + 337355729056326 T^{5} + 643794160690467198 T^{6} + \)\(15\!\cdots\!48\)\( T^{7} + 643794160690467198 p^{3} T^{8} + 337355729056326 p^{6} T^{9} + 1202670029317 p^{9} T^{10} + 491184668 p^{12} T^{11} + 1496388 p^{15} T^{12} + 418 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 - 2635 T + 5424655 T^{2} - 7813480008 T^{3} + 9577271836256 T^{4} - 9607497095580184 T^{5} + 8435216181541044254 T^{6} - \)\(63\!\cdots\!78\)\( T^{7} + 8435216181541044254 p^{3} T^{8} - 9607497095580184 p^{6} T^{9} + 9577271836256 p^{9} T^{10} - 7813480008 p^{12} T^{11} + 5424655 p^{15} T^{12} - 2635 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 - 457 T + 2742314 T^{2} - 1271748743 T^{3} + 3553898601245 T^{4} - 1632895399074508 T^{5} + 2916284527220049784 T^{6} - \)\(12\!\cdots\!44\)\( T^{7} + 2916284527220049784 p^{3} T^{8} - 1632895399074508 p^{6} T^{9} + 3553898601245 p^{9} T^{10} - 1271748743 p^{12} T^{11} + 2742314 p^{15} T^{12} - 457 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 - 592 T + 1073650 T^{2} - 868719106 T^{3} + 1328077884271 T^{4} - 469204225254456 T^{5} + 926071066677623158 T^{6} - \)\(56\!\cdots\!00\)\( T^{7} + 926071066677623158 p^{3} T^{8} - 469204225254456 p^{6} T^{9} + 1328077884271 p^{9} T^{10} - 868719106 p^{12} T^{11} + 1073650 p^{15} T^{12} - 592 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 + 1906 T + 4773643 T^{2} + 5643227298 T^{3} + 8655740565824 T^{4} + 7617605422858678 T^{5} + 9493399876561638956 T^{6} + \)\(72\!\cdots\!12\)\( T^{7} + 9493399876561638956 p^{3} T^{8} + 7617605422858678 p^{6} T^{9} + 8655740565824 p^{9} T^{10} + 5643227298 p^{12} T^{11} + 4773643 p^{15} T^{12} + 1906 p^{18} T^{13} + p^{21} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.68340614886887603538688128090, −6.55840939306120944740231332878, −6.30817627965155909779523464869, −6.23391131525252373794030154958, −5.90033753096494836344236706515, −5.23310383944820812807097873916, −5.19269740561505549037638280512, −5.12993932603101210106068055557, −4.84373148659914741719450906434, −4.75827180136345138551470840975, −4.71704285098030707407427347265, −4.05913429019322440350374346581, −3.93337997471685670618067512399, −3.72752084068713117923020223843, −3.67390495373758240107983465066, −3.64654500629993182497750968523, −3.44316058969392398437979652052, −3.23020095222386699072236285240, −2.84573739504704145327438329099, −2.65515941951464657731438894273, −2.59330988709362911747075112847, −2.06797750947095367109343158962, −2.01057895566647514118964697868, −1.88399783409168388817900905759, −1.74897375990281298245142082470, 0, 0, 0, 0, 0, 0, 0, 1.74897375990281298245142082470, 1.88399783409168388817900905759, 2.01057895566647514118964697868, 2.06797750947095367109343158962, 2.59330988709362911747075112847, 2.65515941951464657731438894273, 2.84573739504704145327438329099, 3.23020095222386699072236285240, 3.44316058969392398437979652052, 3.64654500629993182497750968523, 3.67390495373758240107983465066, 3.72752084068713117923020223843, 3.93337997471685670618067512399, 4.05913429019322440350374346581, 4.71704285098030707407427347265, 4.75827180136345138551470840975, 4.84373148659914741719450906434, 5.12993932603101210106068055557, 5.19269740561505549037638280512, 5.23310383944820812807097873916, 5.90033753096494836344236706515, 6.23391131525252373794030154958, 6.30817627965155909779523464869, 6.55840939306120944740231332878, 6.68340614886887603538688128090
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.