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Properties

Label 2-5e2-5.2-c32-0-41
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  + (3.79e4 + 3.79e4i)2-s + (−3.04e7 + 3.04e7i)3-s − 1.41e9i·4-s − 2.31e12·6-s + (9.26e12 + 9.26e12i)7-s + (2.16e14 − 2.16e14i)8-s + 9.78e11i·9-s + 1.98e16·11-s + (4.29e16 + 4.29e16i)12-s + (8.93e17 − 8.93e17i)13-s + 7.03e17i·14-s + 1.03e19·16-s + (3.66e19 + 3.66e19i)17-s + (−3.71e16 + 3.71e16i)18-s − 2.80e20i·19-s + ⋯
L(s)  = 1  + (0.579 + 0.579i)2-s + (−0.706 + 0.706i)3-s − 0.328i·4-s − 0.819·6-s + (0.278 + 0.278i)7-s + (0.769 − 0.769i)8-s + 0.000527i·9-s + 0.432·11-s + (0.232 + 0.232i)12-s + (1.34 − 1.34i)13-s + 0.323i·14-s + 0.563·16-s + (0.753 + 0.753i)17-s + (−0.000305 + 0.000305i)18-s − 0.972i·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

\(L(\frac{33}{2})\) \(\approx\) \(2.469347296\)
\(L(\frac12)\) \(\approx\) \(2.469347296\)
\(L(17)\) 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 \( 1 + (-3.79e4 - 3.79e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (3.04e7 - 3.04e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (-9.26e12 - 9.26e12i)T + 1.10e27iT^{2} \)
11 \( 1 - 1.98e16T + 2.11e33T^{2} \)
13 \( 1 + (-8.93e17 + 8.93e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (-3.66e19 - 3.66e19i)T + 2.36e39iT^{2} \)
19 \( 1 + 2.80e20iT - 8.31e40T^{2} \)
23 \( 1 + (-2.06e21 + 2.06e21i)T - 3.76e43iT^{2} \)
29 \( 1 - 1.02e23iT - 6.26e46T^{2} \)
31 \( 1 + 9.64e23T + 5.29e47T^{2} \)
37 \( 1 + (7.27e24 + 7.27e24i)T + 1.52e50iT^{2} \)
41 \( 1 - 1.58e25T + 4.06e51T^{2} \)
43 \( 1 + (-9.21e25 + 9.21e25i)T - 1.86e52iT^{2} \)
47 \( 1 + (-3.74e26 - 3.74e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (4.43e27 - 4.43e27i)T - 1.50e55iT^{2} \)
59 \( 1 - 4.27e27iT - 4.64e56T^{2} \)
61 \( 1 + 5.42e28T + 1.35e57T^{2} \)
67 \( 1 + (1.45e29 + 1.45e29i)T + 2.71e58iT^{2} \)
71 \( 1 + 1.17e29T + 1.73e59T^{2} \)
73 \( 1 + (-8.02e28 + 8.02e28i)T - 4.22e59iT^{2} \)
79 \( 1 + 2.05e30iT - 5.29e60T^{2} \)
83 \( 1 + (-4.06e30 + 4.06e30i)T - 2.57e61iT^{2} \)
89 \( 1 + 1.23e31iT - 2.40e62T^{2} \)
97 \( 1 + (2.90e31 + 2.90e31i)T + 3.77e63iT^{2} \)
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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

−10.91880640619204920563168842603, −10.54617783051190816558957069158, −9.027414017017762685204854424078, −7.56299041094919937952338498150, −6.05583264984759569777967136956, −5.53245043335079470088704609972, −4.53654890382237409362344161530, −3.42938289291158297634197024017, −1.55154161739357383218815207092, −0.45058651398358991608555494075, 1.15033357304812157601567381493, 1.74144278310480268068288864950, 3.37219055516864269561474327324, 4.19478572844980274758314337543, 5.58547142808757575640529399574, 6.72687104045288668133173719671, 7.79091691796424235695681091866, 9.197219644785864098984909553981, 10.98132296036543458606980097896, 11.66696982302332473467166458813

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