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Properties

Label 2-5e2-5.3-c32-0-32
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  + (−2.43e4 + 2.43e4i)2-s + (2.67e7 + 2.67e7i)3-s + 3.10e9i·4-s − 1.30e12·6-s + (−4.33e13 + 4.33e13i)7-s + (−1.80e14 − 1.80e14i)8-s − 4.16e14i·9-s + 2.50e16·11-s + (−8.33e16 + 8.33e16i)12-s + (7.52e17 + 7.52e17i)13-s − 2.11e18i·14-s − 4.57e18·16-s + (3.85e19 − 3.85e19i)17-s + (1.01e19 + 1.01e19i)18-s + 1.17e20i·19-s + ⋯
L(s)  = 1  + (−0.371 + 0.371i)2-s + (0.622 + 0.622i)3-s + 0.723i·4-s − 0.462·6-s + (−1.30 + 1.30i)7-s + (−0.640 − 0.640i)8-s − 0.225i·9-s + 0.545·11-s + (−0.450 + 0.450i)12-s + (1.13 + 1.13i)13-s − 0.970i·14-s − 0.247·16-s + (0.791 − 0.791i)17-s + (0.0836 + 0.0836i)18-s + 0.405i·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

\(L(\frac{33}{2})\) \(\approx\) \(1.486894220\)
\(L(\frac12)\) \(\approx\) \(1.486894220\)
\(L(17)\) 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 \( 1 + (2.43e4 - 2.43e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (-2.67e7 - 2.67e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (4.33e13 - 4.33e13i)T - 1.10e27iT^{2} \)
11 \( 1 - 2.50e16T + 2.11e33T^{2} \)
13 \( 1 + (-7.52e17 - 7.52e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (-3.85e19 + 3.85e19i)T - 2.36e39iT^{2} \)
19 \( 1 - 1.17e20iT - 8.31e40T^{2} \)
23 \( 1 + (3.24e21 + 3.24e21i)T + 3.76e43iT^{2} \)
29 \( 1 + 2.76e23iT - 6.26e46T^{2} \)
31 \( 1 + 7.57e23T + 5.29e47T^{2} \)
37 \( 1 + (-6.38e24 + 6.38e24i)T - 1.52e50iT^{2} \)
41 \( 1 + 4.00e25T + 4.06e51T^{2} \)
43 \( 1 + (-2.44e24 - 2.44e24i)T + 1.86e52iT^{2} \)
47 \( 1 + (-7.35e26 + 7.35e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (4.27e26 + 4.27e26i)T + 1.50e55iT^{2} \)
59 \( 1 + 1.79e28iT - 4.64e56T^{2} \)
61 \( 1 + 1.42e28T + 1.35e57T^{2} \)
67 \( 1 + (-1.58e29 + 1.58e29i)T - 2.71e58iT^{2} \)
71 \( 1 - 2.34e29T + 1.73e59T^{2} \)
73 \( 1 + (-1.18e29 - 1.18e29i)T + 4.22e59iT^{2} \)
79 \( 1 + 1.26e30iT - 5.29e60T^{2} \)
83 \( 1 + (7.76e29 + 7.76e29i)T + 2.57e61iT^{2} \)
89 \( 1 - 4.78e30iT - 2.40e62T^{2} \)
97 \( 1 + (1.96e31 - 1.96e31i)T - 3.77e63iT^{2} \)
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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

−11.75479164558686059235789233023, −9.658256928864380617289585084030, −9.183156847642349850549414762476, −8.380995795284224080948307937458, −6.76844236747039043304587388561, −5.95560848622283607058274633861, −3.93740775445052870353176092581, −3.37456159371074066851357859732, −2.27837474765996915036296291353, −0.35828884678756975001035001026, 0.913759327674928722892903881603, 1.39792013054657502836338962277, 2.90726325862829462171582167152, 3.78498442200310625234093085068, 5.65822344655387153967425622991, 6.70480759894145550262492696830, 7.85493009610969128362918352271, 9.088497224791412139477006186222, 10.23604293920114937185472684194, 10.88156329210619794502803961808

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