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Properties

Label 2-325-65.2-c1-0-4
Degree $2$
Conductor $325$
Sign $0.721 + 0.692i$
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  + (−1.27 − 2.20i)2-s + (1.76 + 0.473i)3-s + (−2.24 + 3.88i)4-s + (−1.20 − 4.49i)6-s + (3.08 + 1.77i)7-s + 6.31·8-s + (0.294 + 0.170i)9-s + (−1.44 + 5.39i)11-s + (−5.79 + 5.79i)12-s + (2.82 + 2.23i)13-s − 9.05i·14-s + (−3.56 − 6.17i)16-s + (−0.140 − 0.522i)17-s − 0.865i·18-s + (3.16 − 0.848i)19-s + ⋯
L(s)  = 1  + (−0.900 − 1.55i)2-s + (1.01 + 0.273i)3-s + (−1.12 + 1.94i)4-s + (−0.491 − 1.83i)6-s + (1.16 + 0.672i)7-s + 2.23·8-s + (0.0981 + 0.0566i)9-s + (−0.435 + 1.62i)11-s + (−1.67 + 1.67i)12-s + (0.784 + 0.619i)13-s − 2.42i·14-s + (−0.890 − 1.54i)16-s + (−0.0339 − 0.126i)17-s − 0.204i·18-s + (0.726 − 0.194i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09587 - 0.440995i\)
\(L(\frac12)\) \(\approx\) \(1.09587 - 0.440995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.82 - 2.23i)T \)
good2 \( 1 + (1.27 + 2.20i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.76 - 0.473i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-3.08 - 1.77i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.44 - 5.39i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.140 + 0.522i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.16 + 0.848i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.50 + 5.62i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.795 - 0.459i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.614 - 0.614i)T - 31iT^{2} \)
37 \( 1 + (4.37 - 2.52i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.34 + 1.69i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.48 + 1.46i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + (-1.91 + 1.91i)T - 53iT^{2} \)
59 \( 1 + (-0.0766 - 0.286i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.24 + 2.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.56 - 6.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.36 - 12.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 3.87T + 73T^{2} \)
79 \( 1 + 8.77iT - 79T^{2} \)
83 \( 1 - 5.05iT - 83T^{2} \)
89 \( 1 + (0.609 + 0.163i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.47 + 2.55i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.49748466912735704886739655581, −10.43688317185627128917895538562, −9.657935729477305996575844015600, −8.737326707955402942731924102650, −8.408223778770664844488999598075, −7.25127502107060505371051744220, −4.96919743131427909916771019352, −3.87453951394905928699091549476, −2.55991338521956938906917806849, −1.79183841710739082165622948140, 1.19736808064449843140489169775, 3.41904066032830306370432490794, 5.20027151288881321379352664819, 5.99540272283330389186815228487, 7.47135282692282725994806301589, 7.920708973162713229101579662483, 8.518010655292307103519446642413, 9.292079895512288493110457663589, 10.61540081752330217449087367651, 11.27455982086266021076818808946

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