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Properties

Label 2-791-791.454-c0-0-0
Degree $2$
Conductor $791$
Sign $-0.455 - 0.890i$
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 + (0.0739 + 0.211i)23-s + (0.781 − 0.623i)25-s + (−0.321 − 1.40i)28-s + (0.211 − 1.87i)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 + (0.0739 + 0.211i)23-s + (0.781 − 0.623i)25-s + (−0.321 − 1.40i)28-s + (0.211 − 1.87i)29-s + (0.279 + 1.22i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(791\)    =    \(7 \cdot 113\)
Sign: $-0.455 - 0.890i$
Analytic conductor: \(0.394760\)
Root analytic conductor: \(0.628299\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{791} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 791,\ (\ :0),\ -0.455 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478204029\)
\(L(\frac12)\) \(\approx\) \(1.478204029\)
\(L(1)\) 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 \( 1 + (-0.0739 - 0.211i)T + (-0.781 + 0.623i)T^{2} \)
29 \( 1 + (-0.211 + 1.87i)T + (-0.974 - 0.222i)T^{2} \)
31 \( 1 + (0.623 - 0.781i)T^{2} \)
37 \( 1 + (1.68 + 1.05i)T + (0.433 + 0.900i)T^{2} \)
41 \( 1 + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (-0.656 - 0.0739i)T + (0.974 + 0.222i)T^{2} \)
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 \( 1 + (1.05 - 1.68i)T + (-0.433 - 0.900i)T^{2} \)
71 \( 1 + (0.752 + 0.752i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-1.19 + 0.752i)T + (0.433 - 0.900i)T^{2} \)
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

−10.55063951716698653582970665004, −9.997254748836189969492971630718, −8.807761404520428410748128480978, −7.83211541043492766500715290330, −7.17237212371234499638055511803, −6.34091373240721741945232909768, −5.50373226808591680876250239535, −4.70357377548565316176683958926, −3.72894581259018446457240966906, −2.42034104309377886082509403127, 1.34947849400939360524550330347, 2.92108509027432489732406483477, 3.50967685068487452495231272386, 4.56110137990068450897435291232, 5.43330884259400184128773040385, 6.62507068899348146842996870689, 7.41531312033421723909248314228, 8.830968422716613340248797465777, 9.739705643848415773905445520525, 10.55636753777051131265849647377

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