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Properties

Label 2-325-65.7-c1-0-13
Degree $2$
Conductor $325$
Sign $0.581 + 0.813i$
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  + (0.427 + 0.246i)2-s + (0.243 + 0.908i)3-s + (−0.878 − 1.52i)4-s + (−0.120 + 0.448i)6-s + (−1.83 − 3.18i)7-s − 1.85i·8-s + (1.83 − 1.05i)9-s + (−0.177 − 0.664i)11-s + (1.16 − 1.16i)12-s + (2.92 − 2.11i)13-s − 1.81i·14-s + (−1.29 + 2.24i)16-s + (−2.29 − 0.614i)17-s + 1.04·18-s + (5.29 + 1.41i)19-s + ⋯
L(s)  = 1  + (0.302 + 0.174i)2-s + (0.140 + 0.524i)3-s + (−0.439 − 0.760i)4-s + (−0.0490 + 0.183i)6-s + (−0.694 − 1.20i)7-s − 0.655i·8-s + (0.610 − 0.352i)9-s + (−0.0536 − 0.200i)11-s + (0.337 − 0.337i)12-s + (0.810 − 0.585i)13-s − 0.485i·14-s + (−0.324 + 0.562i)16-s + (−0.556 − 0.149i)17-s + 0.246·18-s + (1.21 + 0.325i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.581 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20794 - 0.620960i\)
\(L(\frac12)\) \(\approx\) \(1.20794 - 0.620960i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.92 + 2.11i)T \)
good2 \( 1 + (-0.427 - 0.246i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.243 - 0.908i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.83 + 3.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.177 + 0.664i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.29 + 0.614i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.29 - 1.41i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.30 - 0.350i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (8.24 + 4.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.81 - 4.81i)T + 31iT^{2} \)
37 \( 1 + (-0.917 + 1.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.534 - 0.143i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.560 + 2.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 3.80T + 47T^{2} \)
53 \( 1 + (-2.47 + 2.47i)T - 53iT^{2} \)
59 \( 1 + (2.69 - 10.0i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.6 - 6.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.73 - 6.47i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 3.37iT - 73T^{2} \)
79 \( 1 - 3.12iT - 79T^{2} \)
83 \( 1 + 2.13T + 83T^{2} \)
89 \( 1 + (-3.26 + 0.874i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.12 - 3.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.21543121016208178353298920190, −10.26312245168278901460571579126, −9.848965089236430535609223631181, −8.913134170170375703964970524100, −7.46741022991755287527303788953, −6.51166930067263998762649831241, −5.46738826280203900016216197689, −4.19116924730912733967428953677, −3.53778010702335621843416175318, −0.954342106547333236551374524965, 2.12185049125853873392080784039, 3.33279679663753754717103605545, 4.59510289801482437736101907364, 5.84255005779189802038537117508, 7.00249680182345525595353330237, 7.979941841605551712037314085082, 8.957069536107144198464628482929, 9.610579217596306882709214693745, 11.14385957345830377419776971200, 11.98679301891509976518963093496

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