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Properties

Label 2-791-791.692-c0-0-0
Degree $2$
Conductor $791$
Sign $0.776 + 0.630i$
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 + (−1.00 − 1.59i)23-s + (0.433 − 0.900i)25-s + (−0.153 + 0.193i)28-s + (1.59 − 0.559i)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 + (−1.00 − 1.59i)23-s + (0.433 − 0.900i)25-s + (−0.153 + 0.193i)28-s + (1.59 − 0.559i)29-s + (0.300 − 0.376i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(791\)    =    \(7 \cdot 113\)
Sign: $0.776 + 0.630i$
Analytic conductor: \(0.394760\)
Root analytic conductor: \(0.628299\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{791} (692, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 791,\ (\ :0),\ 0.776 + 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6577629258\)
\(L(\frac12)\) \(\approx\) \(0.6577629258\)
\(L(1)\) 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 \( 1 + (1.00 + 1.59i)T + (-0.433 + 0.900i)T^{2} \)
29 \( 1 + (-1.59 + 0.559i)T + (0.781 - 0.623i)T^{2} \)
31 \( 1 + (-0.900 + 0.433i)T^{2} \)
37 \( 1 + (0.656 - 0.0739i)T + (0.974 - 0.222i)T^{2} \)
41 \( 1 + (-0.222 + 0.974i)T^{2} \)
43 \( 1 + (-0.351 - 1.00i)T + (-0.781 + 0.623i)T^{2} \)
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 \( 1 + (0.0739 + 0.656i)T + (-0.974 + 0.222i)T^{2} \)
71 \( 1 + (-0.158 + 0.158i)T - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (1.40 + 0.158i)T + (0.974 + 0.222i)T^{2} \)
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

−10.18656537442339539577189489736, −9.858543709221594015631593046863, −8.791391642981190611815175355959, −8.006599838866825626871295188095, −7.04319798434964248550744109285, −6.24819749413245490715457012923, −4.54537667906575317571490669184, −4.36197268719729316617973924403, −2.38309149146194678000824939509, −1.22561629346740439755162228060, 1.27163810891445426735823538129, 3.31841413357012705846915034285, 3.92353714229851649860253388257, 5.52847573253785140060885115290, 6.45663236969282692235132076092, 7.07730357128831184924988229839, 8.336292772182518734618286830381, 8.917975460622163331014454638415, 9.392395132029880091263349324364, 10.23198527725897941524884467237

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