Dirichlet series
| L(s) = 1 | − 8·2-s + 32·4-s − 6·5-s − 10·7-s − 80·8-s + 6·9-s + 48·10-s − 42·11-s + 80·14-s + 120·16-s − 48·18-s − 22·19-s − 192·20-s + 336·22-s + 18·25-s + 24·27-s − 320·28-s + 12·29-s − 32·31-s − 32·32-s + 60·35-s + 192·36-s − 32·37-s + 176·38-s + 480·40-s + 12·41-s − 1.34e3·44-s + ⋯ |
| L(s) = 1 | − 4·2-s + 8·4-s − 6/5·5-s − 1.42·7-s − 10·8-s + 2/3·9-s + 24/5·10-s − 3.81·11-s + 40/7·14-s + 15/2·16-s − 8/3·18-s − 1.15·19-s − 9.59·20-s + 15.2·22-s + 0.719·25-s + 8/9·27-s − 11.4·28-s + 0.413·29-s − 1.03·31-s − 32-s + 12/7·35-s + 16/3·36-s − 0.864·37-s + 4.63·38-s + 12·40-s + 0.292·41-s − 30.5·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.17623\times 10^{7}\) |
| Root analytic conductor: | \(3.03477\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 13^{16} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.05429769509\) |
| \(L(\frac12)\) | \(\approx\) | \(0.05429769509\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p T^{2} )^{4} \) |
| 13 | \( 1 \) | |
| good | 3 | \( ( 1 - p T^{2} - 4 p T^{3} + 32 p T^{4} - 4 p^{3} T^{5} - p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 5 | \( 1 + 6 T + 18 T^{2} + 48 p T^{3} + 89 p T^{4} - 2988 T^{5} + 2862 T^{6} - 36666 T^{7} - 716556 T^{8} - 36666 p^{2} T^{9} + 2862 p^{4} T^{10} - 2988 p^{6} T^{11} + 89 p^{9} T^{12} + 48 p^{11} T^{13} + 18 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \) | |
| 7 | \( 1 + 10 T + 50 T^{2} + 344 T^{3} + 1097 T^{4} - 2932 T^{5} - 25002 T^{6} - 675462 T^{7} - 9431600 T^{8} - 675462 p^{2} T^{9} - 25002 p^{4} T^{10} - 2932 p^{6} T^{11} + 1097 p^{8} T^{12} + 344 p^{10} T^{13} + 50 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 + 42 T + 882 T^{2} + 15408 T^{3} + 260329 T^{4} + 3787308 T^{5} + 48159990 T^{6} + 596208450 T^{7} + 6967811376 T^{8} + 596208450 p^{2} T^{9} + 48159990 p^{4} T^{10} + 3787308 p^{6} T^{11} + 260329 p^{8} T^{12} + 15408 p^{10} T^{13} + 882 p^{12} T^{14} + 42 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 1688 T^{2} + 1278202 T^{4} - 594757616 T^{6} + 197793873595 T^{8} - 594757616 p^{4} T^{10} + 1278202 p^{8} T^{12} - 1688 p^{12} T^{14} + p^{16} T^{16} \) | |
| 19 | \( 1 + 22 T + 242 T^{2} + 9008 T^{3} + 76025 T^{4} - 2432476 T^{5} - 31340490 T^{6} - 1317914226 T^{7} - 53360296016 T^{8} - 1317914226 p^{2} T^{9} - 31340490 p^{4} T^{10} - 2432476 p^{6} T^{11} + 76025 p^{8} T^{12} + 9008 p^{10} T^{13} + 242 p^{12} T^{14} + 22 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 - 2966 T^{2} + 4354945 T^{4} - 4030760174 T^{6} + 2551195667332 T^{8} - 4030760174 p^{4} T^{10} + 4354945 p^{8} T^{12} - 2966 p^{12} T^{14} + p^{16} T^{16} \) | |
| 29 | \( ( 1 - 6 T + 2434 T^{2} - 14688 T^{3} + 2838975 T^{4} - 14688 p^{2} T^{5} + 2434 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 31 | \( 1 + 32 T + 512 T^{2} + 24592 T^{3} + 1881968 T^{4} + 48317872 T^{5} + 884987520 T^{6} + 50977297056 T^{7} + 2940963189598 T^{8} + 50977297056 p^{2} T^{9} + 884987520 p^{4} T^{10} + 48317872 p^{6} T^{11} + 1881968 p^{8} T^{12} + 24592 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 + 32 T + 512 T^{2} + 99640 T^{3} + 1262042 T^{4} - 94176008 T^{5} + 1304267040 T^{6} - 35813533992 T^{7} - 9361847238917 T^{8} - 35813533992 p^{2} T^{9} + 1304267040 p^{4} T^{10} - 94176008 p^{6} T^{11} + 1262042 p^{8} T^{12} + 99640 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 12 T + 72 T^{2} - 90192 T^{3} + 959722 T^{4} + 31710420 T^{5} + 3617673408 T^{6} - 117655860636 T^{7} - 2873533528005 T^{8} - 117655860636 p^{2} T^{9} + 3617673408 p^{4} T^{10} + 31710420 p^{6} T^{11} + 959722 p^{8} T^{12} - 90192 p^{10} T^{13} + 72 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 - 10094 T^{2} + 50078905 T^{4} - 157834972406 T^{6} + 345819050181172 T^{8} - 157834972406 p^{4} T^{10} + 50078905 p^{8} T^{12} - 10094 p^{12} T^{14} + p^{16} T^{16} \) | |
| 47 | \( 1 + 60 T + 1800 T^{2} + 154932 T^{3} + 21951088 T^{4} + 867532140 T^{5} + 24541932312 T^{6} + 2003115348324 T^{7} + 163148147612766 T^{8} + 2003115348324 p^{2} T^{9} + 24541932312 p^{4} T^{10} + 867532140 p^{6} T^{11} + 21951088 p^{8} T^{12} + 154932 p^{10} T^{13} + 1800 p^{12} T^{14} + 60 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( ( 1 + 66 T + 5917 T^{2} + 184818 T^{3} + 12226368 T^{4} + 184818 p^{2} T^{5} + 5917 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 59 | \( 1 + 234 T + 27378 T^{2} + 2355552 T^{3} + 168786385 T^{4} + 10333862388 T^{5} + 571402762614 T^{6} + 29697541439994 T^{7} + 1613332237548864 T^{8} + 29697541439994 p^{2} T^{9} + 571402762614 p^{4} T^{10} + 10333862388 p^{6} T^{11} + 168786385 p^{8} T^{12} + 2355552 p^{10} T^{13} + 27378 p^{12} T^{14} + 234 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( ( 1 + 36 T + 13174 T^{2} + 379800 T^{3} + 70909095 T^{4} + 379800 p^{2} T^{5} + 13174 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 - 14 T + 98 T^{2} + 346544 T^{3} - 3090895 T^{4} - 1619868028 T^{5} + 83027432070 T^{6} + 134866366194 T^{7} - 675706356010592 T^{8} + 134866366194 p^{2} T^{9} + 83027432070 p^{4} T^{10} - 1619868028 p^{6} T^{11} - 3090895 p^{8} T^{12} + 346544 p^{10} T^{13} + 98 p^{12} T^{14} - 14 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 + 162 T + 13122 T^{2} + 965112 T^{3} + 51705241 T^{4} + 2581557804 T^{5} + 205456778118 T^{6} + 19391695489122 T^{7} + 1679565387877200 T^{8} + 19391695489122 p^{2} T^{9} + 205456778118 p^{4} T^{10} + 2581557804 p^{6} T^{11} + 51705241 p^{8} T^{12} + 965112 p^{10} T^{13} + 13122 p^{12} T^{14} + 162 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 + 166 T + 13778 T^{2} + 1402664 T^{3} + 172017437 T^{4} + 14047512068 T^{5} + 945563904750 T^{6} + 1139077533150 p T^{7} + 1353169547476 p^{2} T^{8} + 1139077533150 p^{3} T^{9} + 945563904750 p^{4} T^{10} + 14047512068 p^{6} T^{11} + 172017437 p^{8} T^{12} + 1402664 p^{10} T^{13} + 13778 p^{12} T^{14} + 166 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( ( 1 + 48 T + 10084 T^{2} + 724368 T^{3} + 50280774 T^{4} + 724368 p^{2} T^{5} + 10084 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 83 | \( 1 - 240 T + 28800 T^{2} - 2896512 T^{3} + 298194640 T^{4} - 338245056 p T^{5} + 2344726766592 T^{6} - 177350340390480 T^{7} + 13560551099314782 T^{8} - 177350340390480 p^{2} T^{9} + 2344726766592 p^{4} T^{10} - 338245056 p^{7} T^{11} + 298194640 p^{8} T^{12} - 2896512 p^{10} T^{13} + 28800 p^{12} T^{14} - 240 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 + 210 T + 22050 T^{2} + 1452444 T^{3} + 52717381 T^{4} + 2018092356 T^{5} + 316177930278 T^{6} + 22741338186462 T^{7} + 1650306286515780 T^{8} + 22741338186462 p^{2} T^{9} + 316177930278 p^{4} T^{10} + 2018092356 p^{6} T^{11} + 52717381 p^{8} T^{12} + 1452444 p^{10} T^{13} + 22050 p^{12} T^{14} + 210 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 + 146 T + 10658 T^{2} + 1285708 T^{3} + 255297629 T^{4} + 26103456412 T^{5} + 1916665036902 T^{6} + 221574517764630 T^{7} + 25422626042990164 T^{8} + 221574517764630 p^{2} T^{9} + 1916665036902 p^{4} T^{10} + 26103456412 p^{6} T^{11} + 255297629 p^{8} T^{12} + 1285708 p^{10} T^{13} + 10658 p^{12} T^{14} + 146 p^{14} T^{15} + p^{16} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.75846893570861769244723516343, −4.67489103803409925703871567634, −4.60197926904345379669128226675, −4.55060151144800266118931414659, −4.22742727119266312913704935113, −4.16388648253354861265056838083, −4.15318068146821788839778597451, −3.75417991211067649323650698190, −3.40917787166136553765985999732, −3.22100230225897114873029116215, −3.19232046188150539066113418874, −2.99641085282985579482515762687, −2.82924185211019892478795103183, −2.82046914388160308333554068870, −2.57166464109949680754803680022, −2.49584005566626208371210000692, −1.82893749289420396195999381273, −1.74649189172459735926667940447, −1.71048829300919174848115635425, −1.51088672579828206663170178947, −1.43958788283151208921580899967, −0.53548207358783768230381120645, −0.42845767795437273012637864510, −0.42413464395360098063713175503, −0.22458438779900451450027056059, 0.22458438779900451450027056059, 0.42413464395360098063713175503, 0.42845767795437273012637864510, 0.53548207358783768230381120645, 1.43958788283151208921580899967, 1.51088672579828206663170178947, 1.71048829300919174848115635425, 1.74649189172459735926667940447, 1.82893749289420396195999381273, 2.49584005566626208371210000692, 2.57166464109949680754803680022, 2.82046914388160308333554068870, 2.82924185211019892478795103183, 2.99641085282985579482515762687, 3.19232046188150539066113418874, 3.22100230225897114873029116215, 3.40917787166136553765985999732, 3.75417991211067649323650698190, 4.15318068146821788839778597451, 4.16388648253354861265056838083, 4.22742727119266312913704935113, 4.55060151144800266118931414659, 4.60197926904345379669128226675, 4.67489103803409925703871567634, 4.75846893570861769244723516343
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.