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Properties

Label 2-832-52.35-c0-0-0
Degree 22
Conductor 832832
Sign 0.964+0.265i0.964 + 0.265i
Analytic cond. 0.4152220.415222
Root an. cond. 0.6443770.644377
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  + 5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)45-s + (−0.5 + 0.866i)49-s + 53-s + (−0.5 − 0.866i)61-s + (0.5 − 0.866i)65-s − 73-s + (−0.499 + 0.866i)81-s + (0.5 + 0.866i)85-s + ⋯
L(s)  = 1  + 5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)45-s + (−0.5 + 0.866i)49-s + 53-s + (−0.5 − 0.866i)61-s + (0.5 − 0.866i)65-s − 73-s + (−0.499 + 0.866i)81-s + (0.5 + 0.866i)85-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.964+0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s) L(s)=((0.964+0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.964+0.265i0.964 + 0.265i
Analytic conductor: 0.4152220.415222
Root analytic conductor: 0.6443770.644377
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ832(191,⋅)\chi_{832} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :0), 0.964+0.265i)(2,\ 832,\ (\ :0),\ 0.964 + 0.265i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1282759411.128275941
L(12)L(\frac12) ≈\approx 1.1282759411.128275941
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
13 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good3 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
5 1−T+T2 1 - T + T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
47 1−T2 1 - T^{2}
53 1−T+T2 1 - T + T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
73 1+T+T2 1 + T + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
97 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.34171195898141600290752045253, −9.550728528784383610325028863174, −8.788857535467445963694871471797, −7.960781945023319979958792070121, −6.74967866018315926630608153952, −5.91080782726032255117256853856, −5.38848199087459744727417017856, −3.85995080406162316945946703601, −2.90170479363272988772809371389, −1.45409004826711199249013006458, 1.76348168960243201443293740352, 2.75780103292694502802947627830, 4.18896647589809392843679974444, 5.33245819744144399996348455539, 5.94004397471897637449001737152, 6.99063060318122291659412215519, 7.930126170105022775739781557862, 8.903547049163200532714424180878, 9.605921739771089939772348901243, 10.37266678450107325880979581293

Graph of the ZZ-function along the critical line