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Properties

Label 2-325-13.11-c2-0-40
Degree $2$
Conductor $325$
Sign $0.882 - 0.469i$
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  + (0.963 − 3.59i)2-s + (−1.65 − 2.86i)3-s + (−8.54 − 4.93i)4-s + (−11.9 + 3.19i)6-s + (3.22 + 12.0i)7-s + (−15.4 + 15.4i)8-s + (−0.981 + 1.69i)9-s + (−5.33 − 1.42i)11-s + 32.6i·12-s + (−11.7 + 5.65i)13-s + 46.4·14-s + (20.9 + 36.3i)16-s + (2.81 + 1.62i)17-s + (5.16 + 5.16i)18-s + (−26.6 + 7.14i)19-s + ⋯
L(s)  = 1  + (0.481 − 1.79i)2-s + (−0.551 − 0.955i)3-s + (−2.13 − 1.23i)4-s + (−1.98 + 0.531i)6-s + (0.460 + 1.71i)7-s + (−1.93 + 1.93i)8-s + (−0.109 + 0.188i)9-s + (−0.484 − 0.129i)11-s + 2.72i·12-s + (−0.900 + 0.435i)13-s + 3.31·14-s + (1.31 + 2.27i)16-s + (0.165 + 0.0956i)17-s + (0.287 + 0.287i)18-s + (−1.40 + 0.376i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.882 - 0.469i$
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: \((2,\ 325,\ (\ :1),\ 0.882 - 0.469i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.110506 + 0.0275693i\)
\(L(\frac12)\) \(\approx\) \(0.110506 + 0.0275693i\)
\(L(2)\) 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 \( 1 + (11.7 - 5.65i)T \)
good2 \( 1 + (-0.963 + 3.59i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (1.65 + 2.86i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-3.22 - 12.0i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (5.33 + 1.42i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-2.81 - 1.62i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (26.6 - 7.14i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (8.58 - 4.95i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (26.7 + 46.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (11.1 + 11.1i)T + 961iT^{2} \)
37 \( 1 + (-25.6 - 6.86i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-15.3 + 57.2i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-3.83 - 2.21i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-7.21 + 7.21i)T - 2.20e3iT^{2} \)
53 \( 1 + 36.3T + 2.80e3T^{2} \)
59 \( 1 + (-7.30 - 27.2i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (35.8 - 62.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-12.7 + 47.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-34.5 + 9.24i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (31.2 - 31.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 82.1T + 6.24e3T^{2} \)
83 \( 1 + (-42.9 - 42.9i)T + 6.88e3iT^{2} \)
89 \( 1 + (35.0 + 9.39i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (49.4 - 13.2i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.00617123330944356692180013100, −9.797896397699933988171571614717, −9.000787402419690660721726997541, −7.86949571936724873835240585393, −6.08206965504304754496376990094, −5.41683835905949507210597157781, −4.16936497054444421621710549187, −2.43673227539537236965048550482, −1.90450392520702897800641610111, −0.04728452699289352834526062323, 3.77782657899802614976551311231, 4.65957320206733024377751268684, 5.12567095671258071047421169923, 6.45060215410747607611650059849, 7.43108532115665047933715608040, 7.977787714442100411118664075606, 9.350391501877412776648401913448, 10.35334981171886767598914116711, 11.01063441231194233726586596167, 12.72998806003830716615346417823

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