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Properties

Label 2-325-65.47-c1-0-6
Degree $2$
Conductor $325$
Sign $-0.859 - 0.511i$
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  + 2.38i·2-s + (0.648 − 0.648i)3-s − 3.67·4-s + (1.54 + 1.54i)6-s + 2.38·7-s − 3.99i·8-s + 2.15i·9-s + (−3.88 + 3.88i)11-s + (−2.38 + 2.38i)12-s + (2.76 + 2.31i)13-s + 5.67i·14-s + 2.15·16-s + (0.262 − 0.262i)17-s − 5.14·18-s + (0.587 − 0.587i)19-s + ⋯
L(s)  = 1  + 1.68i·2-s + (0.374 − 0.374i)3-s − 1.83·4-s + (0.630 + 0.630i)6-s + 0.900·7-s − 1.41i·8-s + 0.719i·9-s + (−1.17 + 1.17i)11-s + (−0.687 + 0.687i)12-s + (0.767 + 0.640i)13-s + 1.51i·14-s + 0.539·16-s + (0.0637 − 0.0637i)17-s − 1.21·18-s + (0.134 − 0.134i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363607 + 1.32202i\)
\(L(\frac12)\) \(\approx\) \(0.363607 + 1.32202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.76 - 2.31i)T \)
good2 \( 1 - 2.38iT - 2T^{2} \)
3 \( 1 + (-0.648 + 0.648i)T - 3iT^{2} \)
7 \( 1 - 2.38T + 7T^{2} \)
11 \( 1 + (3.88 - 3.88i)T - 11iT^{2} \)
17 \( 1 + (-0.262 + 0.262i)T - 17iT^{2} \)
19 \( 1 + (-0.587 + 0.587i)T - 19iT^{2} \)
23 \( 1 + (1.98 + 1.98i)T + 23iT^{2} \)
29 \( 1 + 7.92iT - 29T^{2} \)
31 \( 1 + (-6.59 - 6.59i)T + 31iT^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + (6.37 + 6.37i)T + 41iT^{2} \)
43 \( 1 + (3.09 + 3.09i)T + 43iT^{2} \)
47 \( 1 - 5.67T + 47T^{2} \)
53 \( 1 + (1.54 - 1.54i)T - 53iT^{2} \)
59 \( 1 + (5.49 + 5.49i)T + 59iT^{2} \)
61 \( 1 - 4.11T + 61T^{2} \)
67 \( 1 + 3.74iT - 67T^{2} \)
71 \( 1 + (2.04 + 2.04i)T + 71iT^{2} \)
73 \( 1 + 7.56iT - 73T^{2} \)
79 \( 1 - 6.07iT - 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + (-2.23 - 2.23i)T + 89iT^{2} \)
97 \( 1 - 1.88iT - 97T^{2} \)
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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

−12.23591970008009385932557839254, −10.96233686560282629872855126428, −9.861180672368778614916134644234, −8.591264163664044171421527190493, −7.984760960508120713359409772450, −7.39328391129008427505292111459, −6.34897829120056225786662811634, −5.10563690345957862811648053095, −4.48406040009206523928668374998, −2.17612675660854868109676952081, 1.04305375900059686185788337242, 2.76153490479127389737405150286, 3.55854494494467565782556890451, 4.73750625753285089860804945332, 5.96505469661519198638360352628, 8.015045143317535240589149968468, 8.588576513153129580709313276378, 9.670973816203114424676474659536, 10.47940228621361844374764332338, 11.20297142416858273961287780099

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