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Properties

Label 2-325-13.12-c3-0-28
Degree $2$
Conductor $325$
Sign $0.343 - 0.939i$
Analytic cond. $19.1756$
Root an. cond. $4.37899$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.79i·2-s − 3.41·3-s − 14.9·4-s − 16.3i·6-s + 3.51i·7-s − 33.5i·8-s − 15.3·9-s − 25.2i·11-s + 51.1·12-s + (16.0 − 44.0i)13-s − 16.8·14-s + 40.8·16-s + 70.5·17-s − 73.6i·18-s − 3.06i·19-s + ⋯
L(s)  = 1  + 1.69i·2-s − 0.656·3-s − 1.87·4-s − 1.11i·6-s + 0.189i·7-s − 1.48i·8-s − 0.569·9-s − 0.693i·11-s + 1.22·12-s + (0.343 − 0.939i)13-s − 0.321·14-s + 0.637·16-s + 1.00·17-s − 0.964i·18-s − 0.0370i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.343 - 0.939i$
Analytic conductor: \(19.1756\)
Root analytic conductor: \(4.37899\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :3/2),\ 0.343 - 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.012274522\)
\(L(\frac12)\) \(\approx\) \(1.012274522\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.0 + 44.0i)T \)
good2 \( 1 - 4.79iT - 8T^{2} \)
3 \( 1 + 3.41T + 27T^{2} \)
7 \( 1 - 3.51iT - 343T^{2} \)
11 \( 1 + 25.2iT - 1.33e3T^{2} \)
17 \( 1 - 70.5T + 4.91e3T^{2} \)
19 \( 1 + 3.06iT - 6.85e3T^{2} \)
23 \( 1 - 28.9T + 1.21e4T^{2} \)
29 \( 1 + 48.1T + 2.43e4T^{2} \)
31 \( 1 - 96.2iT - 2.97e4T^{2} \)
37 \( 1 + 66.9iT - 5.06e4T^{2} \)
41 \( 1 - 218. iT - 6.89e4T^{2} \)
43 \( 1 + 82.9T + 7.95e4T^{2} \)
47 \( 1 - 114. iT - 1.03e5T^{2} \)
53 \( 1 - 560.T + 1.48e5T^{2} \)
59 \( 1 + 767. iT - 2.05e5T^{2} \)
61 \( 1 + 764.T + 2.26e5T^{2} \)
67 \( 1 - 735. iT - 3.00e5T^{2} \)
71 \( 1 + 628. iT - 3.57e5T^{2} \)
73 \( 1 + 894. iT - 3.89e5T^{2} \)
79 \( 1 - 1.00e3T + 4.93e5T^{2} \)
83 \( 1 - 188. iT - 5.71e5T^{2} \)
89 \( 1 + 1.22e3iT - 7.04e5T^{2} \)
97 \( 1 + 335. iT - 9.12e5T^{2} \)
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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

−11.34031680951990144803453489199, −10.35684341206255155734043643592, −9.060452237505366107002955896503, −8.296350238575034363008235846759, −7.47531638381370386445160797031, −6.25525891467329111208835313981, −5.71083182329536950498193157790, −4.94042291330928002509828865035, −3.28581602008278544963258397652, −0.55067884152213059191524026390, 0.931431637555819560327395187058, 2.25375660420699420187154444743, 3.58137726495194653125893785705, 4.62055855364511024486531516117, 5.75553167636002497833760759015, 7.13773866705135416124884024217, 8.604118663204513785672034065039, 9.491178732308279589925502474452, 10.36488616697287295841080565611, 11.07384038943678100488724745882

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