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Properties

Label 2-325-65.37-c1-0-16
Degree $2$
Conductor $325$
Sign $0.731 + 0.681i$
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  + (1.94 − 1.12i)2-s + (1.91 + 0.514i)3-s + (1.53 − 2.65i)4-s + (4.31 − 1.15i)6-s + (−0.638 + 1.10i)7-s − 2.39i·8-s + (0.820 + 0.473i)9-s + (−5.27 − 1.41i)11-s + (4.30 − 4.30i)12-s + (−0.840 + 3.50i)13-s + 2.87i·14-s + (0.365 + 0.633i)16-s + (−0.833 − 3.11i)17-s + 2.13·18-s + (−0.315 − 1.17i)19-s + ⋯
L(s)  = 1  + (1.37 − 0.795i)2-s + (1.10 + 0.296i)3-s + (0.766 − 1.32i)4-s + (1.76 − 0.472i)6-s + (−0.241 + 0.418i)7-s − 0.848i·8-s + (0.273 + 0.157i)9-s + (−1.59 − 0.426i)11-s + (1.24 − 1.24i)12-s + (−0.233 + 0.972i)13-s + 0.768i·14-s + (0.0913 + 0.158i)16-s + (−0.202 − 0.754i)17-s + 0.502·18-s + (−0.0723 − 0.270i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.731 + 0.681i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.731 + 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.03654 - 1.19577i\)
\(L(\frac12)\) \(\approx\) \(3.03654 - 1.19577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.840 - 3.50i)T \)
good2 \( 1 + (-1.94 + 1.12i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.91 - 0.514i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.638 - 1.10i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.27 + 1.41i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.833 + 3.11i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.315 + 1.17i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.0428 - 0.160i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-8.41 + 4.85i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.233 - 0.233i)T + 31iT^{2} \)
37 \( 1 + (-0.660 - 1.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.129 + 0.483i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.43 + 1.72i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 3.20T + 47T^{2} \)
53 \( 1 + (4.49 - 4.49i)T - 53iT^{2} \)
59 \( 1 + (-0.00222 + 0.000595i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.695 - 1.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.26 - 3.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.7 + 3.14i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 7.34iT - 73T^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 + 2.65T + 83T^{2} \)
89 \( 1 + (1.86 - 6.96i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.62 - 2.09i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.70459128952193785123654751574, −10.77142092283689984923161237013, −9.782080419078193497949773430537, −8.799055239791116857682829892936, −7.80736988156060831824439606910, −6.27446846769061758471028812083, −5.12725187435626550579930660798, −4.18514418659075887036505822220, −2.86518368387796008655304029231, −2.47625961092685262622915562313, 2.57896030191992097732162590801, 3.43361788467524771965915833054, 4.73978228497006714760603841749, 5.68784738374579549423958010432, 6.91738434957095090883652399823, 7.80014728378670504515511846136, 8.334557357667670252868405821557, 9.901215052627527434863116969081, 10.77930374909244207135530854281, 12.47495269210979829342183134788

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