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Properties

Label 2-5e2-5.2-c32-0-5
Degree $2$
Conductor $25$
Sign $0.850 + 0.525i$
Analytic cond. $162.166$
Root an. cond. $12.7344$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.17e4 − 6.17e4i)2-s + (5.69e7 − 5.69e7i)3-s + 3.32e9i·4-s − 7.03e12·6-s + (1.03e13 + 1.03e13i)7-s + (−5.97e13 + 5.97e13i)8-s − 4.63e15i·9-s − 8.55e16·11-s + (1.89e17 + 1.89e17i)12-s + (2.19e16 − 2.19e16i)13-s − 1.27e18i·14-s + 2.16e19·16-s + (−9.87e18 − 9.87e18i)17-s + (−2.86e20 + 2.86e20i)18-s − 1.02e20i·19-s + ⋯
L(s)  = 1  + (−0.942 − 0.942i)2-s + (1.32 − 1.32i)3-s + 0.774i·4-s − 2.49·6-s + (0.311 + 0.311i)7-s + (−0.212 + 0.212i)8-s − 2.50i·9-s − 1.86·11-s + (1.02 + 1.02i)12-s + (0.0329 − 0.0329i)13-s − 0.587i·14-s + 1.17·16-s + (−0.202 − 0.202i)17-s + (−2.35 + 2.35i)18-s − 0.354i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(162.166\)
Root analytic conductor: \(12.7344\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :16),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.6298503448\)
\(L(\frac12)\) \(\approx\) \(0.6298503448\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (6.17e4 + 6.17e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (-5.69e7 + 5.69e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (-1.03e13 - 1.03e13i)T + 1.10e27iT^{2} \)
11 \( 1 + 8.55e16T + 2.11e33T^{2} \)
13 \( 1 + (-2.19e16 + 2.19e16i)T - 4.42e35iT^{2} \)
17 \( 1 + (9.87e18 + 9.87e18i)T + 2.36e39iT^{2} \)
19 \( 1 + 1.02e20iT - 8.31e40T^{2} \)
23 \( 1 + (3.76e21 - 3.76e21i)T - 3.76e43iT^{2} \)
29 \( 1 - 1.14e23iT - 6.26e46T^{2} \)
31 \( 1 + 8.94e23T + 5.29e47T^{2} \)
37 \( 1 + (-2.29e24 - 2.29e24i)T + 1.52e50iT^{2} \)
41 \( 1 + 1.78e25T + 4.06e51T^{2} \)
43 \( 1 + (-4.64e25 + 4.64e25i)T - 1.86e52iT^{2} \)
47 \( 1 + (-2.49e26 - 2.49e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (-7.86e25 + 7.86e25i)T - 1.50e55iT^{2} \)
59 \( 1 + 1.58e28iT - 4.64e56T^{2} \)
61 \( 1 - 2.93e28T + 1.35e57T^{2} \)
67 \( 1 + (-1.50e29 - 1.50e29i)T + 2.71e58iT^{2} \)
71 \( 1 + 4.66e29T + 1.73e59T^{2} \)
73 \( 1 + (-7.79e27 + 7.79e27i)T - 4.22e59iT^{2} \)
79 \( 1 - 4.15e30iT - 5.29e60T^{2} \)
83 \( 1 + (3.04e30 - 3.04e30i)T - 2.57e61iT^{2} \)
89 \( 1 + 2.29e31iT - 2.40e62T^{2} \)
97 \( 1 + (-5.10e31 - 5.10e31i)T + 3.77e63iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.27568948677400363332745113845, −9.937885881632051720969585927897, −8.825473910556537077755907436951, −8.104990361891008871098092955263, −7.26871241778932390402729448215, −5.54731984246417448507379889930, −3.35172557556815614544345535914, −2.43438130924713648358487432752, −1.94798501201660684714391105505, −0.823415751828265212413540653701, 0.15848877215924240235807885253, 2.13224985106805456312457019169, 3.22134430883084306325091357221, 4.38734069387259271743861579566, 5.59985144024193362653233208616, 7.52572214242631879910795991477, 8.094681345692495104053366378255, 8.962955579719342410226409386313, 10.04313006925755791806938796743, 10.67127857585677952397284150901

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