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Properties

Label 2-964-964.955-c0-0-0
Degree $2$
Conductor $964$
Sign $0.995 - 0.0940i$
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 + (−1.14 − 1.58i)5-s + 0.999i·8-s + (0.978 + 0.207i)9-s + (−0.204 − 1.94i)10-s + (1.55 − 1.25i)13-s + (−0.5 + 0.866i)16-s + (−0.0932 + 0.0475i)17-s + (0.743 + 0.669i)18-s + (0.795 − 1.78i)20-s + (−0.873 + 2.68i)25-s + (1.97 − 0.312i)26-s + (−0.244 + 1.14i)29-s + (−0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.14 − 1.58i)5-s + 0.999i·8-s + (0.978 + 0.207i)9-s + (−0.204 − 1.94i)10-s + (1.55 − 1.25i)13-s + (−0.5 + 0.866i)16-s + (−0.0932 + 0.0475i)17-s + (0.743 + 0.669i)18-s + (0.795 − 1.78i)20-s + (−0.873 + 2.68i)25-s + (1.97 − 0.312i)26-s + (−0.244 + 1.14i)29-s + (−0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(964\)    =    \(2^{2} \cdot 241\)
Sign: $0.995 - 0.0940i$
Analytic conductor: \(0.481098\)
Root analytic conductor: \(0.693612\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{964} (955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 964,\ (\ :0),\ 0.995 - 0.0940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.501876515\)
\(L(\frac12)\) \(\approx\) \(1.501876515\)
\(L(1)\) 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 \( 1 + (-0.978 - 0.207i)T^{2} \)
5 \( 1 + (1.14 + 1.58i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.994 - 0.104i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (-1.55 + 1.25i)T + (0.207 - 0.978i)T^{2} \)
17 \( 1 + (0.0932 - 0.0475i)T + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.244 - 1.14i)T + (-0.913 - 0.406i)T^{2} \)
31 \( 1 + (0.994 + 0.104i)T^{2} \)
37 \( 1 + (1.09 + 0.889i)T + (0.207 + 0.978i)T^{2} \)
41 \( 1 + (1.64 + 0.535i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.587 + 0.809i)T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.278i)T + (0.104 + 0.994i)T^{2} \)
59 \( 1 + (-0.978 + 0.207i)T^{2} \)
61 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.104 + 0.994i)T^{2} \)
71 \( 1 + (0.406 - 0.913i)T^{2} \)
73 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.978 - 0.207i)T^{2} \)
89 \( 1 + (1.21 - 0.325i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \)
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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

−10.46571060454620809278042351442, −8.931539423849751624094534249907, −8.446929251194411777173849589639, −7.72095833419313957716385189876, −6.93806055142265768885527604986, −5.58989102627412854871498711748, −5.03202539732887865845355868724, −4.00888639560648924203664576057, −3.50022655149780281047297993266, −1.41189697597664661683715542095, 1.78411530607132770656070691459, 3.17349996782874635615853225140, 3.85983233617268791570060304234, 4.48886512374841149006279199043, 6.15113318898645150646231794465, 6.71120275503030969525574289567, 7.31552387811129758344914332176, 8.481847600127990665557572292185, 9.777808060555848318020218651740, 10.47445693621165555766044772366

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