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Properties

Label 2-791-791.664-c0-0-0
Degree $2$
Conductor $791$
Sign $0.893 + 0.449i$
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  + (0.781 + 0.376i)2-s + (−0.153 − 0.193i)4-s + (−0.222 − 0.974i)7-s + (−0.240 − 1.05i)8-s + (−0.433 − 0.900i)9-s + (1.52 + 1.21i)11-s + (0.193 − 0.846i)14-s + (0.153 − 0.674i)16-s − 0.867i·18-s + (0.734 + 1.52i)22-s + (0.559 − 0.351i)23-s + (−0.433 + 0.900i)25-s + (−0.153 + 0.193i)28-s + (−0.351 − 1.00i)29-s + (−0.300 + 0.376i)32-s + ⋯
L(s)  = 1  + (0.781 + 0.376i)2-s + (−0.153 − 0.193i)4-s + (−0.222 − 0.974i)7-s + (−0.240 − 1.05i)8-s + (−0.433 − 0.900i)9-s + (1.52 + 1.21i)11-s + (0.193 − 0.846i)14-s + (0.153 − 0.674i)16-s − 0.867i·18-s + (0.734 + 1.52i)22-s + (0.559 − 0.351i)23-s + (−0.433 + 0.900i)25-s + (−0.153 + 0.193i)28-s + (−0.351 − 1.00i)29-s + (−0.300 + 0.376i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(791\)    =    \(7 \cdot 113\)
Sign: $0.893 + 0.449i$
Analytic conductor: \(0.394760\)
Root analytic conductor: \(0.628299\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{791} (664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 791,\ (\ :0),\ 0.893 + 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.307837459\)
\(L(\frac12)\) \(\approx\) \(1.307837459\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.222 + 0.974i)T \)
113 \( 1 + (-0.623 - 0.781i)T \)
good2 \( 1 + (-0.781 - 0.376i)T + (0.623 + 0.781i)T^{2} \)
3 \( 1 + (0.433 + 0.900i)T^{2} \)
5 \( 1 + (0.433 - 0.900i)T^{2} \)
11 \( 1 + (-1.52 - 1.21i)T + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.781 - 0.623i)T^{2} \)
19 \( 1 + (-0.433 - 0.900i)T^{2} \)
23 \( 1 + (-0.559 + 0.351i)T + (0.433 - 0.900i)T^{2} \)
29 \( 1 + (0.351 + 1.00i)T + (-0.781 + 0.623i)T^{2} \)
31 \( 1 + (-0.900 + 0.433i)T^{2} \)
37 \( 1 + (-0.211 - 1.87i)T + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + (-0.222 + 0.974i)T^{2} \)
43 \( 1 + (1.59 - 0.559i)T + (0.781 - 0.623i)T^{2} \)
47 \( 1 + (-0.974 - 0.222i)T^{2} \)
53 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 + (-0.433 - 0.900i)T^{2} \)
61 \( 1 + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (-1.87 + 0.211i)T + (0.974 - 0.222i)T^{2} \)
71 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.158 + 1.40i)T + (-0.974 - 0.222i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.781 - 0.623i)T^{2} \)
97 \( 1 + (0.900 + 0.433i)T^{2} \)
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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

−10.18259403668286998719261577881, −9.619102503949491080587159119274, −8.957480349774920830437114858053, −7.50918440468596969129904524344, −6.64100995308081313407993743706, −6.25556824634628790567989405939, −4.89338319177336709193795691589, −4.11906690378275619894366347507, −3.38701033701332887080175762484, −1.27287488453217853658926950575, 2.10051497848160972054234793925, 3.19195062323345409122672725939, 4.00673087578833687576466609694, 5.29564649881141452207005262908, 5.77689209535966988419515693834, 6.93385925187723763809640970362, 8.446258635944279474255132747308, 8.590497607066730792130738957935, 9.597128298056011926388416218370, 10.99100097825657942472581457679

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