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Properties

Label 2-325-65.8-c1-0-5
Degree $2$
Conductor $325$
Sign $-0.209 - 0.977i$
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  + 0.134·2-s + (2.15 + 2.15i)3-s − 1.98·4-s + (0.290 + 0.290i)6-s + 1.90i·7-s − 0.536·8-s + 6.29i·9-s + (−0.290 + 0.290i)11-s + (−4.27 − 4.27i)12-s + (−3.60 − 0.173i)13-s + 0.257i·14-s + 3.89·16-s + (2.53 + 2.53i)17-s + 0.847i·18-s + (3.15 − 3.15i)19-s + ⋯
L(s)  = 1  + 0.0951·2-s + (1.24 + 1.24i)3-s − 0.990·4-s + (0.118 + 0.118i)6-s + 0.721i·7-s − 0.189·8-s + 2.09i·9-s + (−0.0875 + 0.0875i)11-s + (−1.23 − 1.23i)12-s + (−0.998 − 0.0481i)13-s + 0.0687i·14-s + 0.972·16-s + (0.615 + 0.615i)17-s + 0.199i·18-s + (0.723 − 0.723i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.209 - 0.977i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.209 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983393 + 1.21679i\)
\(L(\frac12)\) \(\approx\) \(0.983393 + 1.21679i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 + 0.173i)T \)
good2 \( 1 - 0.134T + 2T^{2} \)
3 \( 1 + (-2.15 - 2.15i)T + 3iT^{2} \)
7 \( 1 - 1.90iT - 7T^{2} \)
11 \( 1 + (0.290 - 0.290i)T - 11iT^{2} \)
17 \( 1 + (-2.53 - 2.53i)T + 17iT^{2} \)
19 \( 1 + (-3.15 + 3.15i)T - 19iT^{2} \)
23 \( 1 + (-2.27 + 2.27i)T - 23iT^{2} \)
29 \( 1 - 2.40iT - 29T^{2} \)
31 \( 1 + (-2.02 - 2.02i)T + 31iT^{2} \)
37 \( 1 + 5.32iT - 37T^{2} \)
41 \( 1 + (1.51 + 1.51i)T + 41iT^{2} \)
43 \( 1 + (0.888 - 0.888i)T - 43iT^{2} \)
47 \( 1 + 6.94iT - 47T^{2} \)
53 \( 1 + (-1.09 - 1.09i)T + 53iT^{2} \)
59 \( 1 + (-8.31 - 8.31i)T + 59iT^{2} \)
61 \( 1 - 7.17T + 61T^{2} \)
67 \( 1 - 0.939T + 67T^{2} \)
71 \( 1 + (7.37 + 7.37i)T + 71iT^{2} \)
73 \( 1 - 6.63T + 73T^{2} \)
79 \( 1 + 4.39iT - 79T^{2} \)
83 \( 1 + 13.4iT - 83T^{2} \)
89 \( 1 + (10.0 + 10.0i)T + 89iT^{2} \)
97 \( 1 - 4.39T + 97T^{2} \)
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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

−11.97869608598787473950582130220, −10.47254818490353238488963740134, −9.852097595899140600732707744652, −9.016749402101977243005837292614, −8.554149990683809367026465052146, −7.45675760768234663407343621329, −5.43642518525346922280754359582, −4.74253598353335533278206201051, −3.63675472769385487684796276623, −2.61159535569169577547504428960, 1.07078487274771431288082181026, 2.81729555775955194771920892294, 3.89388924543594000143747551172, 5.32309529951716967207523713657, 6.85566774281076142133143385258, 7.70902198534757848963095407556, 8.268935770220234714520038828545, 9.463200025607305319477922376978, 9.945236446074148169550070450033, 11.72374577774453000324728652088

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