Properties

Label 1-76-76.35-r1-0-0
Degree 11
Conductor 7676
Sign 0.934+0.356i0.934 + 0.356i
Analytic cond. 8.167338.16733
Root an. cond. 8.167338.16733
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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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

Λ(s)=(76s/2ΓR(s+1)L(s)=((0.934+0.356i)Λ(1s)\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}
Λ(s)=(76s/2ΓR(s+1)L(s)=((0.934+0.356i)Λ(1s)\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: 11
Conductor: 7676    =    22192^{2} \cdot 19
Sign: 0.934+0.356i0.934 + 0.356i
Analytic conductor: 8.167338.16733
Root analytic conductor: 8.167338.16733
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ76(35,)\chi_{76} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 76, (1: ), 0.934+0.356i)(1,\ 76,\ (1:\ ),\ 0.934 + 0.356i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.148157157+0.2114326841i1.148157157 + 0.2114326841i
L(12)L(\frac12) \approx 1.148157157+0.2114326841i1.148157157 + 0.2114326841i
L(1)L(1) \approx 0.8676148607+0.09876783075i0.8676148607 + 0.09876783075i
L(1)L(1) \approx 0.8676148607+0.09876783075i0.8676148607 + 0.09876783075i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
19 1 1
good3 1+(0.766+0.642i)T 1 + (-0.766 + 0.642i)T
5 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
7 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
17 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
23 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
29 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
31 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
37 1+T 1 + T
41 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
43 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
47 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
53 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
59 1+(0.1730.984i)T 1 + (-0.173 - 0.984i)T
61 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
67 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
71 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
73 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
79 1+(0.766+0.642i)T 1 + (-0.766 + 0.642i)T
83 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
89 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
97 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

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

Graph of the ZZ-function along the critical line