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Properties

Label 1-764-764.55-r0-0-0
Degree $1$
Conductor $764$
Sign $-0.0423 - 0.999i$
Analytic cond. $3.54800$
Root an. cond. $3.54800$
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  + (0.879 − 0.475i)3-s + (−0.986 − 0.164i)5-s + 7-s + (0.546 − 0.837i)9-s + (0.789 − 0.614i)11-s + (−0.879 − 0.475i)13-s + (−0.945 + 0.324i)15-s + (−0.677 − 0.735i)17-s + (0.245 + 0.969i)19-s + (0.879 − 0.475i)21-s + (−0.245 − 0.969i)23-s + (0.945 + 0.324i)25-s + (0.0825 − 0.996i)27-s + (−0.945 + 0.324i)29-s + (0.546 − 0.837i)31-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)3-s + (−0.986 − 0.164i)5-s + 7-s + (0.546 − 0.837i)9-s + (0.789 − 0.614i)11-s + (−0.879 − 0.475i)13-s + (−0.945 + 0.324i)15-s + (−0.677 − 0.735i)17-s + (0.245 + 0.969i)19-s + (0.879 − 0.475i)21-s + (−0.245 − 0.969i)23-s + (0.945 + 0.324i)25-s + (0.0825 − 0.996i)27-s + (−0.945 + 0.324i)29-s + (0.546 − 0.837i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0423 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0423 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $-0.0423 - 0.999i$
Analytic conductor: \(3.54800\)
Root analytic conductor: \(3.54800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (0:\ ),\ -0.0423 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.185530462 - 1.236899168i\)
\(L(\frac12)\) \(\approx\) \(1.185530462 - 1.236899168i\)
\(L(1)\) \(\approx\) \(1.216535676 - 0.4724400148i\)
\(L(1)\) \(\approx\) \(1.216535676 - 0.4724400148i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
191 \( 1 \)
good3 \( 1 + (0.879 - 0.475i)T \)
5 \( 1 + (-0.986 - 0.164i)T \)
7 \( 1 + T \)
11 \( 1 + (0.789 - 0.614i)T \)
13 \( 1 + (-0.879 - 0.475i)T \)
17 \( 1 + (-0.677 - 0.735i)T \)
19 \( 1 + (0.245 + 0.969i)T \)
23 \( 1 + (-0.245 - 0.969i)T \)
29 \( 1 + (-0.945 + 0.324i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (-0.546 - 0.837i)T \)
41 \( 1 + (0.0825 + 0.996i)T \)
43 \( 1 + (0.401 + 0.915i)T \)
47 \( 1 + (0.789 - 0.614i)T \)
53 \( 1 + (-0.789 + 0.614i)T \)
59 \( 1 + (-0.546 + 0.837i)T \)
61 \( 1 + (0.677 - 0.735i)T \)
67 \( 1 + (0.677 - 0.735i)T \)
71 \( 1 + (-0.0825 - 0.996i)T \)
73 \( 1 + (-0.789 - 0.614i)T \)
79 \( 1 + (-0.945 - 0.324i)T \)
83 \( 1 + (0.245 - 0.969i)T \)
89 \( 1 + (0.401 - 0.915i)T \)
97 \( 1 + (0.546 + 0.837i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.207333690098055233115098679837, −21.98626708509187525777998964761, −20.805929741862147242021033078199, −20.16453821623059341444276896608, −19.46410538822154185837500658131, −18.9433470764026806411094417896, −17.592394385820463346301496026978, −17.0555434131234755944372927999, −15.655055647681240477571649462795, −15.35395683842627622642141948906, −14.497002166103366325879567735344, −13.94559793507981336566711165077, −12.712139613961441064673824084692, −11.73854861933602627364416277018, −11.1143096544045907059022462160, −10.07810931606382405027319556407, −9.07797512191821601848234710573, −8.44220178180630677025779208803, −7.445693044049918620873516439447, −6.963365847072001325864984397523, −5.157372330063901707846888520916, −4.36000233424267971236719012072, −3.78364791478218211670533341178, −2.50735174180120516220914618504, −1.57081988804824181368597949920, 0.74014555278169394958649712485, 1.937038892113716085030415725475, 3.02589148008278817791825199581, 4.016672034738460843646516408854, 4.78324760541700657894259248405, 6.16936812738232230290813389245, 7.37096349179782046164308402048, 7.82071262479023201118439520303, 8.64448908283224664092283288726, 9.38616689329460665355448953716, 10.68698599269454099969359569251, 11.677190115137367044883486107180, 12.2036518688516436559253268272, 13.17575672207160992399069838897, 14.319084033983240865028074439297, 14.60935722215901597892212634883, 15.501822930923244803079807289141, 16.470436143623240799410029151490, 17.40571437153812837252908153695, 18.43129540290378360405834948768, 18.94347946115917593261989169632, 20.008591849872754512165565010511, 20.25520023910527837473720975679, 21.16757201999851878231167474832, 22.25704440399369030322617084339

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