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Properties

Label 2-5e2-5.3-c32-0-46
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  + (8.66e4 − 8.66e4i)2-s + (−3.32e7 − 3.32e7i)3-s − 1.07e10i·4-s − 5.76e12·6-s + (3.06e13 − 3.06e13i)7-s + (−5.57e14 − 5.57e14i)8-s + 3.55e14i·9-s − 5.53e16·11-s + (−3.56e17 + 3.56e17i)12-s + (−1.61e17 − 1.61e17i)13-s − 5.31e18i·14-s − 5.06e19·16-s + (2.56e19 − 2.56e19i)17-s + (3.08e19 + 3.08e19i)18-s − 4.51e20i·19-s + ⋯
L(s)  = 1  + (1.32 − 1.32i)2-s + (−0.771 − 0.771i)3-s − 2.49i·4-s − 2.04·6-s + (0.922 − 0.922i)7-s + (−1.98 − 1.98i)8-s + 0.191i·9-s − 1.20·11-s + (−1.92 + 1.92i)12-s + (−0.243 − 0.243i)13-s − 2.43i·14-s − 2.74·16-s + (0.527 − 0.527i)17-s + (0.253 + 0.253i)18-s − 1.56i·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

\(L(\frac{33}{2})\) \(\approx\) \(2.748327954\)
\(L(\frac12)\) \(\approx\) \(2.748327954\)
\(L(17)\) 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 \( 1 + (-8.66e4 + 8.66e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (3.32e7 + 3.32e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (-3.06e13 + 3.06e13i)T - 1.10e27iT^{2} \)
11 \( 1 + 5.53e16T + 2.11e33T^{2} \)
13 \( 1 + (1.61e17 + 1.61e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (-2.56e19 + 2.56e19i)T - 2.36e39iT^{2} \)
19 \( 1 + 4.51e20iT - 8.31e40T^{2} \)
23 \( 1 + (2.16e21 + 2.16e21i)T + 3.76e43iT^{2} \)
29 \( 1 + 1.69e23iT - 6.26e46T^{2} \)
31 \( 1 - 6.36e23T + 5.29e47T^{2} \)
37 \( 1 + (1.46e24 - 1.46e24i)T - 1.52e50iT^{2} \)
41 \( 1 + 4.24e25T + 4.06e51T^{2} \)
43 \( 1 + (-1.43e26 - 1.43e26i)T + 1.86e52iT^{2} \)
47 \( 1 + (4.39e26 - 4.39e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (1.94e27 + 1.94e27i)T + 1.50e55iT^{2} \)
59 \( 1 + 2.69e28iT - 4.64e56T^{2} \)
61 \( 1 - 1.09e28T + 1.35e57T^{2} \)
67 \( 1 + (2.17e28 - 2.17e28i)T - 2.71e58iT^{2} \)
71 \( 1 - 5.41e29T + 1.73e59T^{2} \)
73 \( 1 + (8.08e28 + 8.08e28i)T + 4.22e59iT^{2} \)
79 \( 1 + 1.28e30iT - 5.29e60T^{2} \)
83 \( 1 + (-3.95e30 - 3.95e30i)T + 2.57e61iT^{2} \)
89 \( 1 + 6.87e30iT - 2.40e62T^{2} \)
97 \( 1 + (-3.25e31 + 3.25e31i)T - 3.77e63iT^{2} \)
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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

−10.97309743577383327759490239381, −9.893937367778061825866303143164, −7.68173615910902834570295233486, −6.39599744144652379670129977967, −5.15854481484946318881896109807, −4.57620824000756925439361867590, −3.09282143022315122419577783993, −2.06057895832588264581957571515, −0.899251934023661737956220158081, −0.41536985211096624099192529961, 2.15253773940045372140657185441, 3.59886252783691196058214235593, 4.74399813624718509554766335274, 5.38663665408918171997077348382, 5.98398552190110439666193169714, 7.63642976800944150355097948834, 8.399059524824185150949714085339, 10.33587404711992473800620736024, 11.72714743242166931299335823807, 12.50003033823033195564584611846

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