Properties

Label 4-175e2-1.1-c6e2-0-0
Degree $4$
Conductor $30625$
Sign $1$
Analytic cond. $1620.82$
Root an. cond. $6.34503$
Motivic weight $6$
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  + 16·2-s + 64·4-s − 266·7-s − 1.02e3·8-s − 582·9-s + 1.74e3·11-s − 4.25e3·14-s − 1.63e4·16-s − 9.31e3·18-s + 2.79e4·22-s − 9.47e3·23-s − 1.70e4·28-s + 2.22e4·29-s − 6.55e4·32-s − 3.72e4·36-s − 6.00e3·37-s − 6.28e4·43-s + 1.11e5·44-s − 1.51e5·46-s − 4.68e4·49-s + 1.52e5·53-s + 2.72e5·56-s + 3.56e5·58-s + 1.54e5·63-s + 7.86e5·64-s − 9.90e5·67-s − 3.68e5·71-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 0.775·7-s − 2·8-s − 0.798·9-s + 1.31·11-s − 1.55·14-s − 4·16-s − 1.59·18-s + 2.62·22-s − 0.778·23-s − 0.775·28-s + 0.914·29-s − 2·32-s − 0.798·36-s − 0.118·37-s − 0.790·43-s + 1.31·44-s − 1.55·46-s − 0.398·49-s + 1.02·53-s + 1.55·56-s + 1.82·58-s + 0.619·63-s + 3·64-s − 3.29·67-s − 1.03·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30625\)    =    \(5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1620.82\)
Root analytic conductor: \(6.34503\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 30625,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.9349384534\)
\(L(\frac12)\) \(\approx\) \(0.9349384534\)
\(L(4)\) 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 \( 1 \)
7$C_2$ \( 1 + 38 p T + p^{6} T^{2} \)
good2$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2} \)
3$C_2^2$ \( 1 + 194 p T^{2} + p^{12} T^{4} \)
11$C_2$ \( ( 1 - 874 T + p^{6} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4755578 T^{2} + p^{12} T^{4} \)
17$C_2^2$ \( 1 - 748834 p T^{2} + p^{12} T^{4} \)
19$C_2^2$ \( 1 - 84379322 T^{2} + p^{12} T^{4} \)
23$C_2$ \( ( 1 + 206 p T + p^{6} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 11146 T + p^{6} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1020892802 T^{2} + p^{12} T^{4} \)
37$C_2$ \( ( 1 + 3002 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6189133442 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 + 31418 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 16309886018 T^{2} + p^{12} T^{4} \)
53$C_2$ \( ( 1 - 76406 T + p^{6} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 71539567322 T^{2} + p^{12} T^{4} \)
61$C_2^2$ \( 1 - 27356175482 T^{2} + p^{12} T^{4} \)
67$C_2$ \( ( 1 + 495242 T + p^{6} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 184406 T + p^{6} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 298950552578 T^{2} + p^{12} T^{4} \)
79$C_2$ \( ( 1 + 534934 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 142873131578 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 - 597656180162 T^{2} + p^{12} T^{4} \)
97$C_2^2$ \( 1 - 1002631840898 T^{2} + p^{12} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.00959671199722806570489873783, −11.63778276199660885470575384476, −11.40900215205483730793360254123, −10.20522002894314212677244802597, −10.03227319433961134121385262846, −9.065600006469152950582992810981, −8.996211702847818874473736271095, −8.513078668637949097215147131721, −7.60364961117291651816559863889, −6.60895980633511322570869398468, −6.44327273022098385065769748672, −5.84592798493274590707823487794, −5.44483120114792004093139641267, −4.67886938816587593526536255492, −4.16161839509717405407014257039, −3.74442907001165738164534044211, −3.00802983564462121477512168469, −2.71748615879167177521774827043, −1.39987176172518001485823936364, −0.19737064908307279263389476548, 0.19737064908307279263389476548, 1.39987176172518001485823936364, 2.71748615879167177521774827043, 3.00802983564462121477512168469, 3.74442907001165738164534044211, 4.16161839509717405407014257039, 4.67886938816587593526536255492, 5.44483120114792004093139641267, 5.84592798493274590707823487794, 6.44327273022098385065769748672, 6.60895980633511322570869398468, 7.60364961117291651816559863889, 8.513078668637949097215147131721, 8.996211702847818874473736271095, 9.065600006469152950582992810981, 10.03227319433961134121385262846, 10.20522002894314212677244802597, 11.40900215205483730793360254123, 11.63778276199660885470575384476, 12.00959671199722806570489873783

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