Properties

Degree $14$
Conductor $2.258\times 10^{23}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 2·4-s + 8·5-s − 7·7-s − 14·8-s + 32·10-s + 18·11-s − 13-s − 28·14-s − 22·16-s + 2·17-s − 6·19-s + 16·20-s + 72·22-s + 22·23-s + 17·25-s − 4·26-s − 14·28-s + 16·29-s − 18·31-s + 14·32-s + 8·34-s − 56·35-s + 8·37-s − 24·38-s − 112·40-s + 15·41-s + ⋯
L(s)  = 1  + 2.82·2-s + 4-s + 3.57·5-s − 2.64·7-s − 4.94·8-s + 10.1·10-s + 5.42·11-s − 0.277·13-s − 7.48·14-s − 5.5·16-s + 0.485·17-s − 1.37·19-s + 3.57·20-s + 15.3·22-s + 4.58·23-s + 17/5·25-s − 0.784·26-s − 2.64·28-s + 2.97·29-s − 3.23·31-s + 2.47·32-s + 1.37·34-s − 9.46·35-s + 1.31·37-s − 3.89·38-s − 17.7·40-s + 2.34·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 241^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 241^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(3^{14} \cdot 241^{7}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2169} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 3^{14} \cdot 241^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(45.67650278\)
\(L(\frac12)\) \(\approx\) \(45.67650278\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( ( 1 + T )^{7} \)
good2 \( 1 - p^{2} T + 7 p T^{2} - 17 p T^{3} + 37 p T^{4} - 67 p T^{5} + 223 T^{6} - 327 T^{7} + 223 p T^{8} - 67 p^{3} T^{9} + 37 p^{4} T^{10} - 17 p^{5} T^{11} + 7 p^{6} T^{12} - p^{8} T^{13} + p^{7} T^{14} \)
5 \( 1 - 8 T + 47 T^{2} - 38 p T^{3} + 132 p T^{4} - 1907 T^{5} + 5037 T^{6} - 11697 T^{7} + 5037 p T^{8} - 1907 p^{2} T^{9} + 132 p^{4} T^{10} - 38 p^{5} T^{11} + 47 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 + p T + 46 T^{2} + 4 p^{2} T^{3} + 786 T^{4} + 2528 T^{5} + 7897 T^{6} + 21047 T^{7} + 7897 p T^{8} + 2528 p^{2} T^{9} + 786 p^{3} T^{10} + 4 p^{6} T^{11} + 46 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \)
11 \( 1 - 18 T + 194 T^{2} - 1471 T^{3} + 8839 T^{4} - 43563 T^{5} + 182353 T^{6} - 651389 T^{7} + 182353 p T^{8} - 43563 p^{2} T^{9} + 8839 p^{3} T^{10} - 1471 p^{4} T^{11} + 194 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + T + 43 T^{2} + 16 T^{3} + 74 p T^{4} + 171 T^{5} + 1275 p T^{6} + 3431 T^{7} + 1275 p^{2} T^{8} + 171 p^{2} T^{9} + 74 p^{4} T^{10} + 16 p^{4} T^{11} + 43 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 - 2 T + 54 T^{2} - 118 T^{3} + 1511 T^{4} - 4204 T^{5} + 32789 T^{6} - 93345 T^{7} + 32789 p T^{8} - 4204 p^{2} T^{9} + 1511 p^{3} T^{10} - 118 p^{4} T^{11} + 54 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 6 T + 77 T^{2} + 408 T^{3} + 3328 T^{4} + 14851 T^{5} + 91875 T^{6} + 346087 T^{7} + 91875 p T^{8} + 14851 p^{2} T^{9} + 3328 p^{3} T^{10} + 408 p^{4} T^{11} + 77 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 22 T + 329 T^{2} - 3499 T^{3} + 30354 T^{4} - 215473 T^{5} + 1308858 T^{6} - 6746533 T^{7} + 1308858 p T^{8} - 215473 p^{2} T^{9} + 30354 p^{3} T^{10} - 3499 p^{4} T^{11} + 329 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 - 16 T + 228 T^{2} - 1945 T^{3} + 15113 T^{4} - 86708 T^{5} + 503802 T^{6} - 2527253 T^{7} + 503802 p T^{8} - 86708 p^{2} T^{9} + 15113 p^{3} T^{10} - 1945 p^{4} T^{11} + 228 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 18 T + 321 T^{2} + 3457 T^{3} + 35295 T^{4} + 269386 T^{5} + 1944797 T^{6} + 11129437 T^{7} + 1944797 p T^{8} + 269386 p^{2} T^{9} + 35295 p^{3} T^{10} + 3457 p^{4} T^{11} + 321 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 8 T + 140 T^{2} - 446 T^{3} + 5906 T^{4} + 5317 T^{5} + 140628 T^{6} + 725991 T^{7} + 140628 p T^{8} + 5317 p^{2} T^{9} + 5906 p^{3} T^{10} - 446 p^{4} T^{11} + 140 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 - 15 T + 165 T^{2} - 1716 T^{3} + 15349 T^{4} - 112837 T^{5} + 825907 T^{6} - 5652081 T^{7} + 825907 p T^{8} - 112837 p^{2} T^{9} + 15349 p^{3} T^{10} - 1716 p^{4} T^{11} + 165 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 - 14 T + 160 T^{2} - 1152 T^{3} + 8465 T^{4} - 39218 T^{5} + 236797 T^{6} - 1042279 T^{7} + 236797 p T^{8} - 39218 p^{2} T^{9} + 8465 p^{3} T^{10} - 1152 p^{4} T^{11} + 160 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 10 T + 213 T^{2} - 1823 T^{3} + 23033 T^{4} - 162514 T^{5} + 1573229 T^{6} - 9290969 T^{7} + 1573229 p T^{8} - 162514 p^{2} T^{9} + 23033 p^{3} T^{10} - 1823 p^{4} T^{11} + 213 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 15 T + 248 T^{2} + 2459 T^{3} + 26359 T^{4} + 226500 T^{5} + 2031262 T^{6} + 14891311 T^{7} + 2031262 p T^{8} + 226500 p^{2} T^{9} + 26359 p^{3} T^{10} + 2459 p^{4} T^{11} + 248 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 18 T + 283 T^{2} - 3342 T^{3} + 39787 T^{4} - 377400 T^{5} + 3440044 T^{6} - 26583077 T^{7} + 3440044 p T^{8} - 377400 p^{2} T^{9} + 39787 p^{3} T^{10} - 3342 p^{4} T^{11} + 283 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 4 T + 173 T^{2} + 229 T^{3} + 13815 T^{4} + 53354 T^{5} + 1189845 T^{6} + 3012271 T^{7} + 1189845 p T^{8} + 53354 p^{2} T^{9} + 13815 p^{3} T^{10} + 229 p^{4} T^{11} + 173 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - 18 T + 312 T^{2} - 3657 T^{3} + 46177 T^{4} - 442394 T^{5} + 4388546 T^{6} - 34987571 T^{7} + 4388546 p T^{8} - 442394 p^{2} T^{9} + 46177 p^{3} T^{10} - 3657 p^{4} T^{11} + 312 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 50 T + 1452 T^{2} - 29886 T^{3} + 479876 T^{4} - 6256313 T^{5} + 67998420 T^{6} - 622620957 T^{7} + 67998420 p T^{8} - 6256313 p^{2} T^{9} + 479876 p^{3} T^{10} - 29886 p^{4} T^{11} + 1452 p^{5} T^{12} - 50 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 133 T^{2} + 1068 T^{3} + 10948 T^{4} + 119175 T^{5} + 1571649 T^{6} + 6028685 T^{7} + 1571649 p T^{8} + 119175 p^{2} T^{9} + 10948 p^{3} T^{10} + 1068 p^{4} T^{11} + 133 p^{5} T^{12} + p^{7} T^{14} \)
79 \( 1 + 15 T + 468 T^{2} + 5553 T^{3} + 1243 p T^{4} + 946068 T^{5} + 12199714 T^{6} + 95010077 T^{7} + 12199714 p T^{8} + 946068 p^{2} T^{9} + 1243 p^{4} T^{10} + 5553 p^{4} T^{11} + 468 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 - 24 T + 705 T^{2} - 11938 T^{3} + 194891 T^{4} - 2473798 T^{5} + 28249166 T^{6} - 273618813 T^{7} + 28249166 p T^{8} - 2473798 p^{2} T^{9} + 194891 p^{3} T^{10} - 11938 p^{4} T^{11} + 705 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 - 13 T + 260 T^{2} - 2275 T^{3} + 16962 T^{4} - 2186 T^{5} - 807494 T^{6} + 17411725 T^{7} - 807494 p T^{8} - 2186 p^{2} T^{9} + 16962 p^{3} T^{10} - 2275 p^{4} T^{11} + 260 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - T + 396 T^{2} + 357 T^{3} + 75395 T^{4} + 180194 T^{5} + 9562600 T^{6} + 26454385 T^{7} + 9562600 p T^{8} + 180194 p^{2} T^{9} + 75395 p^{3} T^{10} + 357 p^{4} T^{11} + 396 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
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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

−4.08514881967396252029764307237, −4.08509564166718231212101500117, −4.08326628800448549368667722083, −3.91994584278796338296392549665, −3.71598413957196087374871844966, −3.67773855518284955872201957363, −3.52442687753169345961127055598, −3.30910714656706357608688161764, −3.25214284843653085456393215233, −3.18735410594300365675873345132, −3.04668305644489591501918654679, −2.58525179049367731892609542942, −2.51141320784969513110636917914, −2.42690416047087321563416459680, −2.37196706815865171227513388678, −2.27530469017381419724223355674, −1.89916031792756781062024569249, −1.71914014257974484895952677401, −1.58422942899643427705506895548, −1.34530971951818481464123186867, −0.967036410877531381680850765871, −0.966451702440892308446168591401, −0.940626712707947744438495944189, −0.77466463474000457330126862464, −0.26474122094783773564724981735, 0.26474122094783773564724981735, 0.77466463474000457330126862464, 0.940626712707947744438495944189, 0.966451702440892308446168591401, 0.967036410877531381680850765871, 1.34530971951818481464123186867, 1.58422942899643427705506895548, 1.71914014257974484895952677401, 1.89916031792756781062024569249, 2.27530469017381419724223355674, 2.37196706815865171227513388678, 2.42690416047087321563416459680, 2.51141320784969513110636917914, 2.58525179049367731892609542942, 3.04668305644489591501918654679, 3.18735410594300365675873345132, 3.25214284843653085456393215233, 3.30910714656706357608688161764, 3.52442687753169345961127055598, 3.67773855518284955872201957363, 3.71598413957196087374871844966, 3.91994584278796338296392549665, 4.08326628800448549368667722083, 4.08509564166718231212101500117, 4.08514881967396252029764307237

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.