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Properties

Label 2-791-791.734-c0-0-0
Degree $2$
Conductor $791$
Sign $-0.0548 + 0.998i$
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  + (−0.974 + 1.22i)2-s + (−0.321 − 1.40i)4-s + (−0.900 − 0.433i)7-s + (0.626 + 0.301i)8-s + (−0.781 + 0.623i)9-s + (−0.846 − 0.193i)11-s + (1.40 − 0.678i)14-s + (0.321 − 0.154i)16-s − 1.56i·18-s + (1.06 − 0.846i)22-s + (−1.87 − 0.656i)23-s + (−0.781 − 0.623i)25-s + (−0.321 + 1.40i)28-s + (−0.656 + 0.0739i)29-s + (−0.279 + 1.22i)32-s + ⋯
L(s)  = 1  + (−0.974 + 1.22i)2-s + (−0.321 − 1.40i)4-s + (−0.900 − 0.433i)7-s + (0.626 + 0.301i)8-s + (−0.781 + 0.623i)9-s + (−0.846 − 0.193i)11-s + (1.40 − 0.678i)14-s + (0.321 − 0.154i)16-s − 1.56i·18-s + (1.06 − 0.846i)22-s + (−1.87 − 0.656i)23-s + (−0.781 − 0.623i)25-s + (−0.321 + 1.40i)28-s + (−0.656 + 0.0739i)29-s + (−0.279 + 1.22i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0548 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0548 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(791\)    =    \(7 \cdot 113\)
Sign: $-0.0548 + 0.998i$
Analytic conductor: \(0.394760\)
Root analytic conductor: \(0.628299\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{791} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 791,\ (\ :0),\ -0.0548 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04695213019\)
\(L(\frac12)\) \(\approx\) \(0.04695213019\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.900 + 0.433i)T \)
113 \( 1 + (0.222 + 0.974i)T \)
good2 \( 1 + (0.974 - 1.22i)T + (-0.222 - 0.974i)T^{2} \)
3 \( 1 + (0.781 - 0.623i)T^{2} \)
5 \( 1 + (0.781 + 0.623i)T^{2} \)
11 \( 1 + (0.846 + 0.193i)T + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.974 + 0.222i)T^{2} \)
19 \( 1 + (-0.781 + 0.623i)T^{2} \)
23 \( 1 + (1.87 + 0.656i)T + (0.781 + 0.623i)T^{2} \)
29 \( 1 + (0.656 - 0.0739i)T + (0.974 - 0.222i)T^{2} \)
31 \( 1 + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (0.119 + 0.189i)T + (-0.433 + 0.900i)T^{2} \)
41 \( 1 + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.211 + 1.87i)T + (-0.974 + 0.222i)T^{2} \)
47 \( 1 + (-0.433 - 0.900i)T^{2} \)
53 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.781 + 0.623i)T^{2} \)
61 \( 1 + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + (0.189 - 0.119i)T + (0.433 - 0.900i)T^{2} \)
71 \( 1 + (-1.19 - 1.19i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (0.752 - 1.19i)T + (-0.433 - 0.900i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.974 + 0.222i)T^{2} \)
97 \( 1 + (-0.623 + 0.781i)T^{2} \)
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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

−10.17146992993283540510837740925, −9.249653744141135673692752778084, −8.331968606350991413087838640186, −7.83074839484184763393235754397, −6.94411905836462240753551354704, −6.00323749461410470425907712686, −5.45123008946952558210456607224, −3.90552357689907876801769362570, −2.47074174615031804079010987122, −0.06098023281085999802369342617, 1.96756945701567131966954028661, 2.99269709141153601052012192419, 3.77363663012535156037162177639, 5.52935289205358502725394110435, 6.28287149197403900853013504593, 7.71990683484915545943727155277, 8.383182940613063895195685737684, 9.457417163116170557680113415805, 9.679409720723573319674815527181, 10.58633362342678693058237348686

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