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Properties

Label 2-325-65.48-c0-0-1
Degree $2$
Conductor $325$
Sign $0.0235 + 0.999i$
Analytic cond. $0.162196$
Root an. cond. $0.402735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.499 − 0.866i)6-s + (−0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (−0.5 + 0.866i)11-s + (0.707 + 0.707i)13-s + i·14-s + (−0.5 + 0.866i)16-s + (0.965 + 0.258i)17-s + (0.866 − 0.5i)19-s − 21-s + (0.965 + 0.258i)22-s + (−0.965 + 0.258i)23-s + (−0.866 − 0.499i)24-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.499 − 0.866i)6-s + (−0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (−0.5 + 0.866i)11-s + (0.707 + 0.707i)13-s + i·14-s + (−0.5 + 0.866i)16-s + (0.965 + 0.258i)17-s + (0.866 − 0.5i)19-s − 21-s + (0.965 + 0.258i)22-s + (−0.965 + 0.258i)23-s + (−0.866 − 0.499i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.0235 + 0.999i$
Analytic conductor: \(0.162196\)
Root analytic conductor: \(0.402735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :0),\ 0.0235 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8850251812\)
\(L(\frac12)\) \(\approx\) \(0.8850251812\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
7 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
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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

−11.63059497621761545932121579693, −10.56050615984768631866827844337, −9.700662724998107430203580996618, −9.187127538261903639901877672395, −7.906410089059233433079200604910, −6.97596210356435882917856643000, −5.78404614084730037143240795582, −3.85243823344154307243914425099, −2.97143085763957833404969973555, −1.82800012338587602132834440354, 2.87059337366895051463651295716, 3.47397931275578530169078746211, 5.65959708059753166873739952448, 6.12247590666631510814166778547, 7.63210404174625965527707005293, 8.120570882599874155951379439072, 9.072223024032290383025336032861, 9.799432605460163273769131621560, 11.05585196047361038449606580118, 12.14962510755673167376488250744

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