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Properties

Label 2-325-13.12-c1-0-13
Degree $2$
Conductor $325$
Sign $0.435 + 0.899i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
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  − 0.571i·2-s + 0.428·3-s + 1.67·4-s − 0.244i·6-s − 2.67i·7-s − 2.10i·8-s − 2.81·9-s − 5.10i·11-s + 0.715·12-s + (1.57 + 3.24i)13-s − 1.52·14-s + 2.14·16-s + 5.34·17-s + 1.61i·18-s + 6.24i·19-s + ⋯
L(s)  = 1  − 0.404i·2-s + 0.247·3-s + 0.836·4-s − 0.0999i·6-s − 1.01i·7-s − 0.742i·8-s − 0.938·9-s − 1.53i·11-s + 0.206·12-s + (0.435 + 0.899i)13-s − 0.408·14-s + 0.535·16-s + 1.29·17-s + 0.379i·18-s + 1.43i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.435 + 0.899i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.435 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40200 - 0.878651i\)
\(L(\frac12)\) \(\approx\) \(1.40200 - 0.878651i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (-1.57 - 3.24i)T \)
good2 \( 1 + 0.571iT - 2T^{2} \)
3 \( 1 - 0.428T + 3T^{2} \)
7 \( 1 + 2.67iT - 7T^{2} \)
11 \( 1 + 5.10iT - 11T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 - 6.24iT - 19T^{2} \)
23 \( 1 + 2.42T + 23T^{2} \)
29 \( 1 - 2.67T + 29T^{2} \)
31 \( 1 + 0.244iT - 31T^{2} \)
37 \( 1 - 3.32iT - 37T^{2} \)
41 \( 1 - 6.48iT - 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 - 2.67iT - 47T^{2} \)
53 \( 1 - 4.20T + 53T^{2} \)
59 \( 1 - 0.899iT - 59T^{2} \)
61 \( 1 + 5.81T + 61T^{2} \)
67 \( 1 + 2.18iT - 67T^{2} \)
71 \( 1 - 6.24iT - 71T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 + 3.63T + 79T^{2} \)
83 \( 1 + 9.81iT - 83T^{2} \)
89 \( 1 + 7.63iT - 89T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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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

−11.50244674623486724429202756859, −10.59922150076698400097495287630, −9.871014579669331020419477797972, −8.459396131578689769845415387278, −7.77898313594428322246098196564, −6.48293003178000099590288477798, −5.73951147936088571792146869453, −3.80407580979924953548347026866, −3.08714308887926826851126581818, −1.31453243815159125238767852348, 2.16597279139550307894027083046, 3.13866581739674056490207426959, 5.12649220157358248639883320298, 5.86020772513960203313180208091, 7.00701819562183120238788364056, 7.943692419710875875259196048989, 8.777578116326118340474513018473, 9.896007771289026476052111960619, 10.89778673091568584246420513129, 11.97045595201104751691246878524

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