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Properties

Label 2-5e2-5.2-c32-0-19
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  + (5.10e3 + 5.10e3i)2-s + (7.18e6 − 7.18e6i)3-s − 4.24e9i·4-s + 7.32e10·6-s + (−2.17e13 − 2.17e13i)7-s + (4.35e13 − 4.35e13i)8-s + 1.74e15i·9-s − 5.19e16·11-s + (−3.04e16 − 3.04e16i)12-s + (−6.04e17 + 6.04e17i)13-s − 2.21e17i·14-s − 1.77e19·16-s + (−2.55e19 − 2.55e19i)17-s + (−8.92e18 + 8.92e18i)18-s + 1.62e20i·19-s + ⋯
L(s)  = 1  + (0.0778 + 0.0778i)2-s + (0.166 − 0.166i)3-s − 0.987i·4-s + 0.0259·6-s + (−0.653 − 0.653i)7-s + (0.154 − 0.154i)8-s + 0.944i·9-s − 1.13·11-s + (−0.164 − 0.164i)12-s + (−0.907 + 0.907i)13-s − 0.101i·14-s − 0.963·16-s + (−0.524 − 0.524i)17-s + (−0.0734 + 0.0734i)18-s + 0.564i·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

\(L(\frac{33}{2})\) \(\approx\) \(1.238622979\)
\(L(\frac12)\) \(\approx\) \(1.238622979\)
\(L(17)\) 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 \( 1 + (-5.10e3 - 5.10e3i)T + 4.29e9iT^{2} \)
3 \( 1 + (-7.18e6 + 7.18e6i)T - 1.85e15iT^{2} \)
7 \( 1 + (2.17e13 + 2.17e13i)T + 1.10e27iT^{2} \)
11 \( 1 + 5.19e16T + 2.11e33T^{2} \)
13 \( 1 + (6.04e17 - 6.04e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (2.55e19 + 2.55e19i)T + 2.36e39iT^{2} \)
19 \( 1 - 1.62e20iT - 8.31e40T^{2} \)
23 \( 1 + (-2.79e21 + 2.79e21i)T - 3.76e43iT^{2} \)
29 \( 1 - 3.52e23iT - 6.26e46T^{2} \)
31 \( 1 - 4.98e23T + 5.29e47T^{2} \)
37 \( 1 + (1.32e25 + 1.32e25i)T + 1.52e50iT^{2} \)
41 \( 1 - 1.17e26T + 4.06e51T^{2} \)
43 \( 1 + (9.57e25 - 9.57e25i)T - 1.86e52iT^{2} \)
47 \( 1 + (-4.63e26 - 4.63e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (-7.08e26 + 7.08e26i)T - 1.50e55iT^{2} \)
59 \( 1 + 1.33e28iT - 4.64e56T^{2} \)
61 \( 1 - 2.13e28T + 1.35e57T^{2} \)
67 \( 1 + (2.29e29 + 2.29e29i)T + 2.71e58iT^{2} \)
71 \( 1 + 3.37e28T + 1.73e59T^{2} \)
73 \( 1 + (-4.11e29 + 4.11e29i)T - 4.22e59iT^{2} \)
79 \( 1 + 9.94e29iT - 5.29e60T^{2} \)
83 \( 1 + (1.82e30 - 1.82e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 3.30e30iT - 2.40e62T^{2} \)
97 \( 1 + (5.20e31 + 5.20e31i)T + 3.77e63iT^{2} \)
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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

−10.95408748446180784553300656256, −10.25637336477940989993930618630, −9.149644329373776984560766669790, −7.55765454581069179825759627673, −6.70017999851874117081414448104, −5.29596846661285206431657747384, −4.47922137000016122640761981190, −2.74535290235071290635798665971, −1.80541340500541205501868617668, −0.46358968772587633378856620701, 0.44332001933605909268905000901, 2.51513381719513873330123481657, 2.97618688233608155666363377598, 4.23035803942703150685093552206, 5.57926784465036324053929855093, 6.92660880953766219387142685489, 8.080428729308975779736094121802, 9.099974338393898934048952817983, 10.24336190647747390727355376427, 11.78140632510399183381714349589

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