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Properties

Label 2-925-37.36-c1-0-28
Degree $2$
Conductor $925$
Sign $0.635 - 0.772i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.694i·2-s + 2.37·3-s + 1.51·4-s + 1.65i·6-s − 2.16·7-s + 2.44i·8-s + 2.65·9-s + 4.35·11-s + 3.60·12-s + 5.72i·13-s − 1.50i·14-s + 1.33·16-s − 6.00i·17-s + 1.84i·18-s − 4.14i·19-s + ⋯
L(s)  = 1  + 0.491i·2-s + 1.37·3-s + 0.758·4-s + 0.674i·6-s − 0.819·7-s + 0.863i·8-s + 0.883·9-s + 1.31·11-s + 1.04·12-s + 1.58i·13-s − 0.402i·14-s + 0.334·16-s − 1.45i·17-s + 0.434i·18-s − 0.949i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.635 - 0.772i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (776, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.635 - 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65744 + 1.25518i\)
\(L(\frac12)\) \(\approx\) \(2.65744 + 1.25518i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (-3.86 + 4.69i)T \)
good2 \( 1 - 0.694iT - 2T^{2} \)
3 \( 1 - 2.37T + 3T^{2} \)
7 \( 1 + 2.16T + 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
13 \( 1 - 5.72iT - 13T^{2} \)
17 \( 1 + 6.00iT - 17T^{2} \)
19 \( 1 + 4.14iT - 19T^{2} \)
23 \( 1 + 0.165iT - 23T^{2} \)
29 \( 1 - 8.05iT - 29T^{2} \)
31 \( 1 - 0.0682iT - 31T^{2} \)
41 \( 1 + 8.59T + 41T^{2} \)
43 \( 1 + 0.445iT - 43T^{2} \)
47 \( 1 - 6.94T + 47T^{2} \)
53 \( 1 + 4.63T + 53T^{2} \)
59 \( 1 + 9.44iT - 59T^{2} \)
61 \( 1 - 2.38iT - 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 2.32T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + 8.06iT - 79T^{2} \)
83 \( 1 - 6.59T + 83T^{2} \)
89 \( 1 - 2.34iT - 89T^{2} \)
97 \( 1 + 8.58iT - 97T^{2} \)
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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

−9.774337943307778606954633728672, −9.094390588679512694301147632780, −8.775552609628562915837796333689, −7.41653458872379829751517946918, −6.94427537048570884775643164355, −6.30768143184635465936493853751, −4.82806250816532720163384490056, −3.61521148609048080228748540956, −2.79704696214330284463343883335, −1.76160382124581900592082915442, 1.39651574162145039145466779484, 2.55334910312654963193152856429, 3.45662124346519935860078030983, 3.91161784420135905812576664031, 5.93086241408256700713250327373, 6.48667677173668525117961486239, 7.71507419509066876385897262266, 8.211742036859965150019790396272, 9.236658576414630026351234096792, 10.01377330735470334832995296824

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