Properties

Label 2-791-791.307-c0-0-0
Degree $2$
Conductor $791$
Sign $0.419 + 0.907i$
Analytic cond. $0.394760$
Root an. cond. $0.628299$
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  + (−0.433 − 1.90i)2-s + (−2.52 + 1.21i)4-s + (0.623 + 0.781i)7-s + (2.19 + 2.74i)8-s + (0.974 + 0.222i)9-s + (−0.678 + 1.40i)11-s + (1.21 − 1.52i)14-s + (2.52 − 3.16i)16-s − 1.94i·18-s + (2.97 + 0.678i)22-s + (0.189 + 1.68i)23-s + (0.974 − 0.222i)25-s + (−2.52 − 1.21i)28-s + (−1.68 − 1.05i)29-s + (−3.94 − 1.90i)32-s + ⋯
L(s)  = 1  + (−0.433 − 1.90i)2-s + (−2.52 + 1.21i)4-s + (0.623 + 0.781i)7-s + (2.19 + 2.74i)8-s + (0.974 + 0.222i)9-s + (−0.678 + 1.40i)11-s + (1.21 − 1.52i)14-s + (2.52 − 3.16i)16-s − 1.94i·18-s + (2.97 + 0.678i)22-s + (0.189 + 1.68i)23-s + (0.974 − 0.222i)25-s + (−2.52 − 1.21i)28-s + (−1.68 − 1.05i)29-s + (−3.94 − 1.90i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(791\)    =    \(7 \cdot 113\)
Sign: $0.419 + 0.907i$
Analytic conductor: \(0.394760\)
Root analytic conductor: \(0.628299\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{791} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 791,\ (\ :0),\ 0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7162986499\)
\(L(\frac12)\) \(\approx\) \(0.7162986499\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.623 - 0.781i)T \)
113 \( 1 + (0.900 - 0.433i)T \)
good2 \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \)
3 \( 1 + (-0.974 - 0.222i)T^{2} \)
5 \( 1 + (-0.974 + 0.222i)T^{2} \)
11 \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.433 - 0.900i)T^{2} \)
19 \( 1 + (0.974 + 0.222i)T^{2} \)
23 \( 1 + (-0.189 - 1.68i)T + (-0.974 + 0.222i)T^{2} \)
29 \( 1 + (1.68 + 1.05i)T + (0.433 + 0.900i)T^{2} \)
31 \( 1 + (-0.222 + 0.974i)T^{2} \)
37 \( 1 + (0.351 + 1.00i)T + (-0.781 + 0.623i)T^{2} \)
41 \( 1 + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.119 + 0.189i)T + (-0.433 - 0.900i)T^{2} \)
47 \( 1 + (-0.781 - 0.623i)T^{2} \)
53 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.974 + 0.222i)T^{2} \)
61 \( 1 + (0.623 + 0.781i)T^{2} \)
67 \( 1 + (-1.00 + 0.351i)T + (0.781 - 0.623i)T^{2} \)
71 \( 1 + (-1.33 + 1.33i)T - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-0.467 + 1.33i)T + (-0.781 - 0.623i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.433 - 0.900i)T^{2} \)
97 \( 1 + (0.222 + 0.974i)T^{2} \)
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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

−10.42266377987154448775725372965, −9.553164912230101076147312413869, −9.192288816216981559426067959566, −7.926489688398056592330754426070, −7.43481383720023319535156842428, −5.32561004763828664496565354451, −4.66351574668139823973008693054, −3.66975026455840992415552781272, −2.29018704340616531511241849411, −1.69724643302410716621258612141, 1.00171147182685974879987293577, 3.71980327707765271699251803048, 4.74777289983312382205070453882, 5.44359679047190599574061759909, 6.60378464918598083310774430048, 7.10104770411726318090203153909, 8.065062227165433533190741431518, 8.542724215194936189636475298718, 9.496004047213947895953582282017, 10.46328094487734294432843375578

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