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Properties

Label 6-5e3-1.1-c17e3-0-0
Degree $6$
Conductor $125$
Sign $1$
Analytic cond. $768.853$
Root an. cond. $3.02673$
Motivic weight $17$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 118·2-s + 1.59e4·3-s − 1.81e5·4-s + 1.17e6·5-s + 1.88e6·6-s + 2.13e6·7-s − 5.39e6·8-s + 9.53e7·9-s + 1.38e8·10-s + 1.78e9·11-s − 2.88e9·12-s + 5.41e9·13-s + 2.52e8·14-s + 1.86e10·15-s + 1.37e10·16-s − 2.74e10·17-s + 1.12e10·18-s − 2.99e10·19-s − 2.12e11·20-s + 3.41e10·21-s + 2.11e11·22-s + 1.62e10·23-s − 8.60e10·24-s + 9.15e11·25-s + 6.38e11·26-s + 5.26e11·27-s − 3.87e11·28-s + ⋯
L(s)  = 1  + 0.325·2-s + 1.40·3-s − 1.38·4-s + 1.34·5-s + 0.457·6-s + 0.140·7-s − 0.113·8-s + 0.738·9-s + 0.437·10-s + 2.51·11-s − 1.93·12-s + 1.84·13-s + 0.0457·14-s + 1.88·15-s + 0.799·16-s − 0.952·17-s + 0.240·18-s − 0.404·19-s − 1.85·20-s + 0.196·21-s + 0.820·22-s + 0.0432·23-s − 0.159·24-s + 6/5·25-s + 0.600·26-s + 0.358·27-s − 0.193·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+17/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $1$
Analytic conductor: \(768.853\)
Root analytic conductor: \(3.02673\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 125,\ (\ :17/2, 17/2, 17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(7.831544225\)
\(L(\frac12)\) \(\approx\) \(7.831544225\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$ \( ( 1 - p^{8} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 59 p T + 24389 p^{3} T^{2} - 304715 p^{7} T^{3} + 24389 p^{20} T^{4} - 59 p^{35} T^{5} + p^{51} T^{6} \)
3$S_4\times C_2$ \( 1 - 15944 T + 1961353 p^{4} T^{2} - 19006636720 p^{4} T^{3} + 1961353 p^{21} T^{4} - 15944 p^{34} T^{5} + p^{51} T^{6} \)
7$S_4\times C_2$ \( 1 - 2139308 T + 77842072434051 p T^{2} - 1280732219764989400 p^{3} T^{3} + 77842072434051 p^{18} T^{4} - 2139308 p^{34} T^{5} + p^{51} T^{6} \)
11$S_4\times C_2$ \( 1 - 1789747516 T + 211635823457168915 p T^{2} - \)\(15\!\cdots\!20\)\( p^{2} T^{3} + 211635823457168915 p^{18} T^{4} - 1789747516 p^{34} T^{5} + p^{51} T^{6} \)
13$S_4\times C_2$ \( 1 - 32039626 p^{2} T + 183847518947561147 p^{2} T^{2} - \)\(43\!\cdots\!20\)\( p^{3} T^{3} + 183847518947561147 p^{19} T^{4} - 32039626 p^{36} T^{5} + p^{51} T^{6} \)
17$S_4\times C_2$ \( 1 + 27402303962 T + \)\(15\!\cdots\!47\)\( T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!47\)\( p^{17} T^{4} + 27402303962 p^{34} T^{5} + p^{51} T^{6} \)
19$S_4\times C_2$ \( 1 + 29956565300 T + \)\(53\!\cdots\!17\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!17\)\( p^{17} T^{4} + 29956565300 p^{34} T^{5} + p^{51} T^{6} \)
23$S_4\times C_2$ \( 1 - 16254077844 T + \)\(73\!\cdots\!93\)\( T^{2} - \)\(92\!\cdots\!60\)\( T^{3} + \)\(73\!\cdots\!93\)\( p^{17} T^{4} - 16254077844 p^{34} T^{5} + p^{51} T^{6} \)
29$S_4\times C_2$ \( 1 + 2528278831750 T + \)\(13\!\cdots\!27\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!27\)\( p^{17} T^{4} + 2528278831750 p^{34} T^{5} + p^{51} T^{6} \)
31$S_4\times C_2$ \( 1 + 521256054664 T + \)\(32\!\cdots\!65\)\( T^{2} - \)\(60\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!65\)\( p^{17} T^{4} + 521256054664 p^{34} T^{5} + p^{51} T^{6} \)
37$S_4\times C_2$ \( 1 - 31762746900498 T + \)\(16\!\cdots\!27\)\( T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!27\)\( p^{17} T^{4} - 31762746900498 p^{34} T^{5} + p^{51} T^{6} \)
41$S_4\times C_2$ \( 1 - 86833482954446 T + \)\(91\!\cdots\!15\)\( T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!15\)\( p^{17} T^{4} - 86833482954446 p^{34} T^{5} + p^{51} T^{6} \)
43$S_4\times C_2$ \( 1 - 89258046385744 T + \)\(12\!\cdots\!93\)\( T^{2} - \)\(91\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!93\)\( p^{17} T^{4} - 89258046385744 p^{34} T^{5} + p^{51} T^{6} \)
47$S_4\times C_2$ \( 1 - 348182738140228 T + \)\(56\!\cdots\!17\)\( T^{2} - \)\(68\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!17\)\( p^{17} T^{4} - 348182738140228 p^{34} T^{5} + p^{51} T^{6} \)
53$S_4\times C_2$ \( 1 - 44014499212594 T + \)\(31\!\cdots\!43\)\( T^{2} + \)\(39\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!43\)\( p^{17} T^{4} - 44014499212594 p^{34} T^{5} + p^{51} T^{6} \)
59$S_4\times C_2$ \( 1 + 2133293278957100 T + \)\(47\!\cdots\!57\)\( T^{2} + \)\(55\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!57\)\( p^{17} T^{4} + 2133293278957100 p^{34} T^{5} + p^{51} T^{6} \)
61$S_4\times C_2$ \( 1 - 2822990449991866 T + \)\(58\!\cdots\!15\)\( T^{2} - \)\(90\!\cdots\!20\)\( T^{3} + \)\(58\!\cdots\!15\)\( p^{17} T^{4} - 2822990449991866 p^{34} T^{5} + p^{51} T^{6} \)
67$S_4\times C_2$ \( 1 - 374173462156688 T + \)\(17\!\cdots\!97\)\( T^{2} - \)\(29\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!97\)\( p^{17} T^{4} - 374173462156688 p^{34} T^{5} + p^{51} T^{6} \)
71$S_4\times C_2$ \( 1 + 9029705707562224 T + \)\(11\!\cdots\!65\)\( T^{2} + \)\(55\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!65\)\( p^{17} T^{4} + 9029705707562224 p^{34} T^{5} + p^{51} T^{6} \)
73$S_4\times C_2$ \( 1 + 5906564941861106 T + \)\(13\!\cdots\!43\)\( T^{2} + \)\(51\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!43\)\( p^{17} T^{4} + 5906564941861106 p^{34} T^{5} + p^{51} T^{6} \)
79$S_4\times C_2$ \( 1 - 9183397142621600 T + \)\(41\!\cdots\!77\)\( T^{2} - \)\(32\!\cdots\!00\)\( T^{3} + \)\(41\!\cdots\!77\)\( p^{17} T^{4} - 9183397142621600 p^{34} T^{5} + p^{51} T^{6} \)
83$S_4\times C_2$ \( 1 - 15992201117651544 T + \)\(41\!\cdots\!93\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(41\!\cdots\!93\)\( p^{17} T^{4} - 15992201117651544 p^{34} T^{5} + p^{51} T^{6} \)
89$S_4\times C_2$ \( 1 + 31272661736951250 T + \)\(35\!\cdots\!87\)\( T^{2} + \)\(68\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!87\)\( p^{17} T^{4} + 31272661736951250 p^{34} T^{5} + p^{51} T^{6} \)
97$S_4\times C_2$ \( 1 - 28207632381796278 T + \)\(16\!\cdots\!67\)\( T^{2} - \)\(28\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!67\)\( p^{17} T^{4} - 28207632381796278 p^{34} T^{5} + p^{51} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.52760301754205548641352522724, −16.97184868396499349281845568665, −16.36333002850223249810572669030, −15.53797070347130403206545765411, −14.43850499953354706739244332423, −14.42265921724215901422088248354, −14.18050414684220991636889894747, −13.36796168767200125607593882349, −13.25419151005070291404008829589, −12.63575763431364271797657075103, −11.36426146000042134445895139680, −10.88429768527273124802254675646, −9.744181791137771270821176932431, −9.096276102271981844063281453576, −9.007357013575578198400944637326, −8.639285122035680608624750600428, −7.45826875672673278899382571089, −6.23251696957918084760928191908, −6.01966647911643016894700672962, −4.54354435880039083086726394021, −4.09724958081888545131008704702, −3.49955013173593770312773893330, −2.29980625798811169293843238338, −1.48746059537090573453486945999, −0.900878131228105644880959382096, 0.900878131228105644880959382096, 1.48746059537090573453486945999, 2.29980625798811169293843238338, 3.49955013173593770312773893330, 4.09724958081888545131008704702, 4.54354435880039083086726394021, 6.01966647911643016894700672962, 6.23251696957918084760928191908, 7.45826875672673278899382571089, 8.639285122035680608624750600428, 9.007357013575578198400944637326, 9.096276102271981844063281453576, 9.744181791137771270821176932431, 10.88429768527273124802254675646, 11.36426146000042134445895139680, 12.63575763431364271797657075103, 13.25419151005070291404008829589, 13.36796168767200125607593882349, 14.18050414684220991636889894747, 14.42265921724215901422088248354, 14.43850499953354706739244332423, 15.53797070347130403206545765411, 16.36333002850223249810572669030, 16.97184868396499349281845568665, 17.52760301754205548641352522724

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