Properties

Degree $6$
Conductor $15625$
Sign $-1$
Motivic weight $9$
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 33·2-s − 89·3-s − 53·4-s + 2.93e3·6-s − 5.25e3·7-s + 1.93e4·8-s + 3.55e3·9-s − 5.46e4·11-s + 4.71e3·12-s − 2.15e5·13-s + 1.73e5·14-s − 5.02e5·16-s − 3.34e5·17-s − 1.17e5·18-s + 8.18e5·19-s + 4.67e5·21-s + 1.80e6·22-s − 3.52e6·23-s − 1.72e6·24-s + 7.12e6·26-s − 1.70e6·27-s + 2.78e5·28-s + 2.17e6·29-s + 4.27e6·31-s + 6.15e6·32-s + 4.86e6·33-s + 1.10e7·34-s + ⋯
L(s)  = 1  − 1.45·2-s − 0.634·3-s − 0.103·4-s + 0.925·6-s − 0.827·7-s + 1.67·8-s + 0.180·9-s − 1.12·11-s + 0.0656·12-s − 2.09·13-s + 1.20·14-s − 1.91·16-s − 0.972·17-s − 0.263·18-s + 1.44·19-s + 0.525·21-s + 1.64·22-s − 2.62·23-s − 1.06·24-s + 3.05·26-s − 0.618·27-s + 0.0856·28-s + 0.571·29-s + 0.831·31-s + 1.03·32-s + 0.714·33-s + 1.41·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(15625\)    =    \(5^{6}\)
Sign: $-1$
Motivic weight: \(9\)
Character: induced by $\chi_{25} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 15625,\ (\ :9/2, 9/2, 9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 \)
good2$S_4\times C_2$ \( 1 + 33 T + 571 p T^{2} + 2505 p^{3} T^{3} + 571 p^{10} T^{4} + 33 p^{18} T^{5} + p^{27} T^{6} \)
3$S_4\times C_2$ \( 1 + 89 T + 1456 p T^{2} + 197885 p^{2} T^{3} + 1456 p^{10} T^{4} + 89 p^{18} T^{5} + p^{27} T^{6} \)
7$S_4\times C_2$ \( 1 + 5258 T + 11082951 p T^{2} + 9176976100 p^{2} T^{3} + 11082951 p^{10} T^{4} + 5258 p^{18} T^{5} + p^{27} T^{6} \)
11$S_4\times C_2$ \( 1 + 54699 T + 5816187440 T^{2} + 182671409832855 T^{3} + 5816187440 p^{9} T^{4} + 54699 p^{18} T^{5} + p^{27} T^{6} \)
13$S_4\times C_2$ \( 1 + 215884 T + 31225156343 T^{2} + 3094983572287480 T^{3} + 31225156343 p^{9} T^{4} + 215884 p^{18} T^{5} + p^{27} T^{6} \)
17$S_4\times C_2$ \( 1 + 334983 T + 355030662062 T^{2} + 77675997792847395 T^{3} + 355030662062 p^{9} T^{4} + 334983 p^{18} T^{5} + p^{27} T^{6} \)
19$S_4\times C_2$ \( 1 - 818845 T + 468173656712 T^{2} - 302339390932836385 T^{3} + 468173656712 p^{9} T^{4} - 818845 p^{18} T^{5} + p^{27} T^{6} \)
23$S_4\times C_2$ \( 1 + 3526854 T + 8700596724993 T^{2} + 13086552252942250620 T^{3} + 8700596724993 p^{9} T^{4} + 3526854 p^{18} T^{5} + p^{27} T^{6} \)
29$S_4\times C_2$ \( 1 - 2175480 T + 1119231270883 p T^{2} - 59155485309271560240 T^{3} + 1119231270883 p^{10} T^{4} - 2175480 p^{18} T^{5} + p^{27} T^{6} \)
31$S_4\times C_2$ \( 1 - 4274066 T + 82851493809465 T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + 82851493809465 p^{9} T^{4} - 4274066 p^{18} T^{5} + p^{27} T^{6} \)
37$S_4\times C_2$ \( 1 - 10305042 T + 340842196290747 T^{2} - \)\(25\!\cdots\!40\)\( T^{3} + 340842196290747 p^{9} T^{4} - 10305042 p^{18} T^{5} + p^{27} T^{6} \)
41$S_4\times C_2$ \( 1 - 5926311 T + 232043727124790 T^{2} - \)\(23\!\cdots\!95\)\( T^{3} + 232043727124790 p^{9} T^{4} - 5926311 p^{18} T^{5} + p^{27} T^{6} \)
43$S_4\times C_2$ \( 1 - 24429956 T + 1676050268074193 T^{2} - \)\(57\!\cdots\!00\)\( p T^{3} + 1676050268074193 p^{9} T^{4} - 24429956 p^{18} T^{5} + p^{27} T^{6} \)
47$S_4\times C_2$ \( 1 + 66858708 T + 4517266130065277 T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + 4517266130065277 p^{9} T^{4} + 66858708 p^{18} T^{5} + p^{27} T^{6} \)
53$S_4\times C_2$ \( 1 + 132620514 T + 9209248487675243 T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + 9209248487675243 p^{9} T^{4} + 132620514 p^{18} T^{5} + p^{27} T^{6} \)
59$S_4\times C_2$ \( 1 - 5670960 T + 19337695182532817 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + 19337695182532817 p^{9} T^{4} - 5670960 p^{18} T^{5} + p^{27} T^{6} \)
61$S_4\times C_2$ \( 1 - 125306926 T + 22192130203358915 T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + 22192130203358915 p^{9} T^{4} - 125306926 p^{18} T^{5} + p^{27} T^{6} \)
67$S_4\times C_2$ \( 1 + 88829483 T + 30562384919894712 T^{2} + \)\(76\!\cdots\!95\)\( T^{3} + 30562384919894712 p^{9} T^{4} + 88829483 p^{18} T^{5} + p^{27} T^{6} \)
71$S_4\times C_2$ \( 1 - 297550596 T + 87036096018332165 T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + 87036096018332165 p^{9} T^{4} - 297550596 p^{18} T^{5} + p^{27} T^{6} \)
73$S_4\times C_2$ \( 1 + 181321729 T + 94365252526618118 T^{2} + \)\(75\!\cdots\!45\)\( T^{3} + 94365252526618118 p^{9} T^{4} + 181321729 p^{18} T^{5} + p^{27} T^{6} \)
79$S_4\times C_2$ \( 1 + 310025170 T + 176092553119892457 T^{2} + \)\(48\!\cdots\!60\)\( T^{3} + 176092553119892457 p^{9} T^{4} + 310025170 p^{18} T^{5} + p^{27} T^{6} \)
83$S_4\times C_2$ \( 1 - 731088801 T + 685433348047337568 T^{2} - \)\(27\!\cdots\!65\)\( T^{3} + 685433348047337568 p^{9} T^{4} - 731088801 p^{18} T^{5} + p^{27} T^{6} \)
89$S_4\times C_2$ \( 1 + 1103860035 T + 1320664213408319502 T^{2} + \)\(75\!\cdots\!55\)\( T^{3} + 1320664213408319502 p^{9} T^{4} + 1103860035 p^{18} T^{5} + p^{27} T^{6} \)
97$S_4\times C_2$ \( 1 - 332236842 T + 1487745026246658927 T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + 1487745026246658927 p^{9} T^{4} - 332236842 p^{18} T^{5} + p^{27} T^{6} \)
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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

−14.54192802842692284546358285298, −13.75364134509268459491960891098, −13.70917626106507298691176968364, −13.16619608604816373812186690775, −12.55770890990143471509376138485, −12.35701020613872017876767637838, −11.68169259467473820630641620589, −11.30244727891195735897360890196, −10.73460499143510124342817246010, −10.00152887582417565495041228708, −9.855387815386657612873864730795, −9.475566648017787305391727405541, −9.441403860475069735441949996622, −8.319240754853765611889985219103, −7.903825817238894517680062799071, −7.84078528087851967200430637627, −6.93609597030698285625299696812, −6.48045144075158179156089249643, −5.73105075948872739414964611968, −4.94605313112059336701637850812, −4.69893143914022749985463743303, −3.95119921526953394778238225382, −2.76087902285144678126819856350, −2.29197459251356216679271108537, −1.23343417415332397762648624664, 0, 0, 0, 1.23343417415332397762648624664, 2.29197459251356216679271108537, 2.76087902285144678126819856350, 3.95119921526953394778238225382, 4.69893143914022749985463743303, 4.94605313112059336701637850812, 5.73105075948872739414964611968, 6.48045144075158179156089249643, 6.93609597030698285625299696812, 7.84078528087851967200430637627, 7.903825817238894517680062799071, 8.319240754853765611889985219103, 9.441403860475069735441949996622, 9.475566648017787305391727405541, 9.855387815386657612873864730795, 10.00152887582417565495041228708, 10.73460499143510124342817246010, 11.30244727891195735897360890196, 11.68169259467473820630641620589, 12.35701020613872017876767637838, 12.55770890990143471509376138485, 13.16619608604816373812186690775, 13.70917626106507298691176968364, 13.75364134509268459491960891098, 14.54192802842692284546358285298

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