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Properties

Label 2-325-13.11-c2-0-33
Degree $2$
Conductor $325$
Sign $-0.972 - 0.231i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.590 − 2.20i)2-s + (−1.05 − 1.83i)3-s + (−1.03 − 0.598i)4-s + (−4.66 + 1.24i)6-s + (0.314 + 1.17i)7-s + (4.51 − 4.51i)8-s + (2.26 − 3.91i)9-s + (−13.3 − 3.57i)11-s + 2.53i·12-s + (−12.5 − 3.48i)13-s + 2.76·14-s + (−9.67 − 16.7i)16-s + (−8.20 − 4.73i)17-s + (−7.28 − 7.28i)18-s + (23.4 − 6.28i)19-s + ⋯
L(s)  = 1  + (0.295 − 1.10i)2-s + (−0.352 − 0.610i)3-s + (−0.259 − 0.149i)4-s + (−0.776 + 0.208i)6-s + (0.0449 + 0.167i)7-s + (0.564 − 0.564i)8-s + (0.251 − 0.435i)9-s + (−1.21 − 0.324i)11-s + 0.211i·12-s + (−0.963 − 0.268i)13-s + 0.197·14-s + (−0.604 − 1.04i)16-s + (−0.482 − 0.278i)17-s + (−0.404 − 0.404i)18-s + (1.23 − 0.330i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.972 - 0.231i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.972 - 0.231i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.162483 + 1.38757i\)
\(L(\frac12)\) \(\approx\) \(0.162483 + 1.38757i\)
\(L(2)\) 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 \( 1 + (12.5 + 3.48i)T \)
good2 \( 1 + (-0.590 + 2.20i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (1.05 + 1.83i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-0.314 - 1.17i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (13.3 + 3.57i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (8.20 + 4.73i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-23.4 + 6.28i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (10.0 - 5.80i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (2.30 + 3.99i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (21.8 + 21.8i)T + 961iT^{2} \)
37 \( 1 + (-19.1 - 5.14i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (14.6 - 54.8i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-61.4 - 35.4i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-22.8 + 22.8i)T - 2.20e3iT^{2} \)
53 \( 1 - 3.82T + 2.80e3T^{2} \)
59 \( 1 + (17.8 + 66.5i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-20.0 + 34.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-15.0 + 56.1i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (53.0 - 14.2i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (34.0 - 34.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 27.0T + 6.24e3T^{2} \)
83 \( 1 + (-20.0 - 20.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (30.1 + 8.08i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-172. + 46.1i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.25512205644303018069209696771, −10.11569986894157873656072853139, −9.431944256089322771904512881117, −7.77171802560066978799749819273, −7.17521762847314798950602226957, −5.80913163615327496521793752079, −4.65801161632364156848912493066, −3.21635134041014868226800770042, −2.17826831171222763526328895343, −0.59705601398274392973930195881, 2.25648764646788072246581741017, 4.24630833822757534130125575840, 5.13371054776843287442290311168, 5.78089908867508402253053676810, 7.38074865351809647246198694646, 7.51401208753056304187497843303, 8.980518718334900031663636464747, 10.28762846149830929238598755423, 10.64866439779062192577872967359, 11.86443698613773716855116443021

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