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Properties

Label 2-325-13.12-c3-0-2
Degree $2$
Conductor $325$
Sign $-0.159 + 0.987i$
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  + 2.90i·2-s − 8.97·3-s − 0.433·4-s − 26.0i·6-s + 4.22i·7-s + 21.9i·8-s + 53.6·9-s + 6.14i·11-s + 3.89·12-s + (−7.45 + 46.2i)13-s − 12.2·14-s − 67.2·16-s − 75.8·17-s + 155. i·18-s + 123. i·19-s + ⋯
L(s)  = 1  + 1.02i·2-s − 1.72·3-s − 0.0541·4-s − 1.77i·6-s + 0.228i·7-s + 0.971i·8-s + 1.98·9-s + 0.168i·11-s + 0.0936·12-s + (−0.159 + 0.987i)13-s − 0.234·14-s − 1.05·16-s − 1.08·17-s + 2.03i·18-s + 1.48i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.159 + 0.987i$
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: \((2,\ 325,\ (\ :3/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3023532205\)
\(L(\frac12)\) \(\approx\) \(0.3023532205\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.45 - 46.2i)T \)
good2 \( 1 - 2.90iT - 8T^{2} \)
3 \( 1 + 8.97T + 27T^{2} \)
7 \( 1 - 4.22iT - 343T^{2} \)
11 \( 1 - 6.14iT - 1.33e3T^{2} \)
17 \( 1 + 75.8T + 4.91e3T^{2} \)
19 \( 1 - 123. iT - 6.85e3T^{2} \)
23 \( 1 - 112.T + 1.21e4T^{2} \)
29 \( 1 + 81.0T + 2.43e4T^{2} \)
31 \( 1 + 259. iT - 2.97e4T^{2} \)
37 \( 1 + 142. iT - 5.06e4T^{2} \)
41 \( 1 - 198. iT - 6.89e4T^{2} \)
43 \( 1 + 346.T + 7.95e4T^{2} \)
47 \( 1 + 343. iT - 1.03e5T^{2} \)
53 \( 1 - 24.8T + 1.48e5T^{2} \)
59 \( 1 + 247. iT - 2.05e5T^{2} \)
61 \( 1 + 524.T + 2.26e5T^{2} \)
67 \( 1 - 43.6iT - 3.00e5T^{2} \)
71 \( 1 + 737. iT - 3.57e5T^{2} \)
73 \( 1 - 344. iT - 3.89e5T^{2} \)
79 \( 1 + 1.02e3T + 4.93e5T^{2} \)
83 \( 1 + 954. iT - 5.71e5T^{2} \)
89 \( 1 - 1.24e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.74e3iT - 9.12e5T^{2} \)
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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

−11.60682644843678804553290747926, −11.22046693979093244781846417511, −10.14794681933301041920555958392, −8.954201670637935469431819546151, −7.61681650686695785431837926957, −6.75513036991496119213435309326, −6.11001008489814136811696304654, −5.28137131148912593846492890918, −4.30531514396970760244754386138, −1.86328212799798738883524333056, 0.14772497708305181015331053056, 1.21130415737024820904107957474, 2.91195419256871161244084208866, 4.43739059453834811985438066505, 5.34949366898951645869796212813, 6.60585220628857752583455442458, 7.16577707309408348325326569301, 8.979746240808345561520123000315, 10.16074630052369258555166356957, 10.84951654475361572998909846420

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