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Properties

Label 2-964-964.123-c0-0-0
Degree $2$
Conductor $964$
Sign $0.766 + 0.641i$
Analytic cond. $0.481098$
Root an. cond. $0.693612$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (1.73 + 0.564i)5-s − 0.999i·8-s + (−0.913 − 0.406i)9-s + (1.78 − 0.379i)10-s + (−1.30 + 0.846i)13-s + (−0.5 − 0.866i)16-s + (−1.24 + 0.196i)17-s + (−0.994 + 0.104i)18-s + (1.35 − 1.22i)20-s + (1.89 + 1.37i)25-s + (−0.705 + 1.38i)26-s + (−0.773 + 1.73i)29-s + (−0.866 − 0.499i)32-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (1.73 + 0.564i)5-s − 0.999i·8-s + (−0.913 − 0.406i)9-s + (1.78 − 0.379i)10-s + (−1.30 + 0.846i)13-s + (−0.5 − 0.866i)16-s + (−1.24 + 0.196i)17-s + (−0.994 + 0.104i)18-s + (1.35 − 1.22i)20-s + (1.89 + 1.37i)25-s + (−0.705 + 1.38i)26-s + (−0.773 + 1.73i)29-s + (−0.866 − 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(964\)    =    \(2^{2} \cdot 241\)
Sign: $0.766 + 0.641i$
Analytic conductor: \(0.481098\)
Root analytic conductor: \(0.693612\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{964} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 964,\ (\ :0),\ 0.766 + 0.641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.870676786\)
\(L(\frac12)\) \(\approx\) \(1.870676786\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
241 \( 1 + T \)
good3 \( 1 + (0.913 + 0.406i)T^{2} \)
5 \( 1 + (-1.73 - 0.564i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.207 + 0.978i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (1.30 - 0.846i)T + (0.406 - 0.913i)T^{2} \)
17 \( 1 + (1.24 - 0.196i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.587 + 0.809i)T^{2} \)
29 \( 1 + (0.773 - 1.73i)T + (-0.669 - 0.743i)T^{2} \)
31 \( 1 + (0.207 - 0.978i)T^{2} \)
37 \( 1 + (1.18 + 0.770i)T + (0.406 + 0.913i)T^{2} \)
41 \( 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.951 + 0.309i)T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 0.0850i)T + (0.978 - 0.207i)T^{2} \)
59 \( 1 + (0.913 - 0.406i)T^{2} \)
61 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.978 - 0.207i)T^{2} \)
71 \( 1 + (-0.743 + 0.669i)T^{2} \)
73 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.913 + 0.406i)T^{2} \)
89 \( 1 + (-1.05 - 0.281i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
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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

−10.37093802354308410746851084943, −9.280179753013271888140341342541, −9.057059109359447729379857371004, −6.99597314628818949708958727821, −6.67257957281826496164604984074, −5.58846782516866845053378188363, −5.13017202962019847145708585413, −3.71715368195988098166075543624, −2.48028870742304082874394113766, −1.99392308413627855383003056863, 2.20449974343680745449640673403, 2.73399257016781634698182435891, 4.48800158093065254019396955514, 5.26154401620597865834847227108, 5.81806406005809754280091707573, 6.61608701159997578633306455650, 7.73348903345505821603812106595, 8.602484057177998702699807106440, 9.430904196764383297255171370931, 10.25962145477614537431157637517

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