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Properties

Label 1-76-76.63-r1-0-0
Degree $1$
Conductor $76$
Sign $0.934 - 0.356i$
Analytic cond. $8.16733$
Root an. cond. $8.16733$
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  + (−0.766 − 0.642i)3-s + (−0.939 + 0.342i)5-s + (0.5 + 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (0.766 − 0.642i)13-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)17-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (0.766 − 0.642i)25-s + (0.5 − 0.866i)27-s + (0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)3-s + (−0.939 + 0.342i)5-s + (0.5 + 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (0.766 − 0.642i)13-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)17-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (0.766 − 0.642i)25-s + (0.5 − 0.866i)27-s + (0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(8.16733\)
Root analytic conductor: \(8.16733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (1:\ ),\ 0.934 - 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.148157157 - 0.2114326841i\)
\(L(\frac12)\) \(\approx\) \(1.148157157 - 0.2114326841i\)
\(L(1)\) \(\approx\) \(0.8676148607 - 0.09876783075i\)
\(L(1)\) \(\approx\) \(0.8676148607 - 0.09876783075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−31.02544744042214713503034126765, −30.182066149791190715714281211411, −28.65651303857500874783077260545, −27.90562624615299958718850772673, −27.037518674585718210271648808525, −26.0739892010995034155241545905, −24.2809133098292607279212407255, −23.36007206378984271210225504865, −22.7265847324990504714160466115, −21.15108321058282310200389396560, −20.39028686174776523166758120900, −19.07760865713355978964521072640, −17.47330386225900830376748478146, −16.747191215700704870723607526383, −15.59071313475156347994425673886, −14.55852683448197249779297403463, −12.7727413000849082368981001, −11.56787792574267677478893483013, −10.75738862509840503331399201720, −9.308543883691556608727010600657, −7.7799814129823128520259854436, −6.384310696984467435155758517794, −4.5515419878775381291061844324, −3.96684770177517138241835959944, −1.01246517932589726062569630817, 0.939438472503433382520684821743, 3.05368430117884304326361197924, 4.98973681323677219047174459796, 6.26459546746393040983536759633, 7.60701607963533984727180810232, 8.73737990887081527816779126448, 10.92163356345042863745789642039, 11.551822875062796260420107085654, 12.61491864920623724039135807847, 14.0952197505415757648601532766, 15.50870441980884039778224570457, 16.48312804299469141872363742489, 18.00386863522102067357605082740, 18.676067695795702969205874820229, 19.735150566869649122795302251, 21.37886098097510563727661994936, 22.53070479712262387213849618618, 23.35494972075371236401471432577, 24.442609528907593227545999370532, 25.302328357425522166650528943672, 27.167698467132921824395217681316, 27.66892946024838474915347156961, 28.87251269441061825606024124974, 30.04066833236684085432472264954, 30.84096285299500936196508163316

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