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Properties

Label 2-5e2-5.3-c32-0-27
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  + (7.09e4 − 7.09e4i)2-s + (4.83e7 + 4.83e7i)3-s − 5.77e9i·4-s + 6.86e12·6-s + (−2.09e13 + 2.09e13i)7-s + (−1.05e14 − 1.05e14i)8-s + 2.82e15i·9-s + 6.47e16·11-s + (2.79e17 − 2.79e17i)12-s + (4.20e17 + 4.20e17i)13-s + 2.97e18i·14-s + 9.89e18·16-s + (1.82e19 − 1.82e19i)17-s + (2.00e20 + 2.00e20i)18-s − 1.23e20i·19-s + ⋯
L(s)  = 1  + (1.08 − 1.08i)2-s + (1.12 + 1.12i)3-s − 1.34i·4-s + 2.43·6-s + (−0.631 + 0.631i)7-s + (−0.373 − 0.373i)8-s + 1.52i·9-s + 1.40·11-s + (1.51 − 1.51i)12-s + (0.631 + 0.631i)13-s + 1.36i·14-s + 0.536·16-s + (0.374 − 0.374i)17-s + (1.65 + 1.65i)18-s − 0.428i·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: $\chi_{25} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :16),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(7.104378608\)
\(L(\frac12)\) \(\approx\) \(7.104378608\)
\(L(17)\) 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 \( 1 + (-7.09e4 + 7.09e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (-4.83e7 - 4.83e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (2.09e13 - 2.09e13i)T - 1.10e27iT^{2} \)
11 \( 1 - 6.47e16T + 2.11e33T^{2} \)
13 \( 1 + (-4.20e17 - 4.20e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (-1.82e19 + 1.82e19i)T - 2.36e39iT^{2} \)
19 \( 1 + 1.23e20iT - 8.31e40T^{2} \)
23 \( 1 + (6.48e21 + 6.48e21i)T + 3.76e43iT^{2} \)
29 \( 1 - 4.68e23iT - 6.26e46T^{2} \)
31 \( 1 - 2.94e22T + 5.29e47T^{2} \)
37 \( 1 + (1.50e25 - 1.50e25i)T - 1.52e50iT^{2} \)
41 \( 1 - 4.08e25T + 4.06e51T^{2} \)
43 \( 1 + (5.01e25 + 5.01e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (4.23e26 - 4.23e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (-2.94e27 - 2.94e27i)T + 1.50e55iT^{2} \)
59 \( 1 + 8.62e27iT - 4.64e56T^{2} \)
61 \( 1 - 1.77e28T + 1.35e57T^{2} \)
67 \( 1 + (-2.48e28 + 2.48e28i)T - 2.71e58iT^{2} \)
71 \( 1 - 3.63e29T + 1.73e59T^{2} \)
73 \( 1 + (-5.94e29 - 5.94e29i)T + 4.22e59iT^{2} \)
79 \( 1 - 1.45e30iT - 5.29e60T^{2} \)
83 \( 1 + (3.27e30 + 3.27e30i)T + 2.57e61iT^{2} \)
89 \( 1 + 8.42e29iT - 2.40e62T^{2} \)
97 \( 1 + (-3.56e30 + 3.56e30i)T - 3.77e63iT^{2} \)
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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

−11.62097756500654733382511926935, −10.39194417528989350933077982300, −9.376996210166399212325550928633, −8.587726833976154226384651041966, −6.45286941086959965321082428976, −4.96813405700094652279043370267, −3.97863679973885887996507447896, −3.36918486957195270676581807533, −2.48763606228167728976729800809, −1.36790263554516058295865097219, 0.77076604621807493413740146782, 1.86460351287257915333077667166, 3.54935844558190582594626079104, 3.82775312835483697251944868889, 5.83675396387455547881353743897, 6.62073378576540111746859297060, 7.51680838320976005530346161478, 8.362837973110683121894961060839, 9.832189433313400747600679256016, 11.99455061719049672799824401458

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