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Properties

Label 2-825-11.5-c1-0-17
Degree 22
Conductor 825825
Sign 0.634−0.772i0.634 - 0.772i
Analytic cond. 6.587656.58765
Root an. cond. 2.566642.56664
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.392 + 1.20i)2-s + (−0.809 − 0.587i)3-s + (0.311 − 0.226i)4-s + (0.392 − 1.20i)6-s + (−0.390 + 0.284i)7-s + (2.45 + 1.78i)8-s + (0.309 + 0.951i)9-s + (−0.982 − 3.16i)11-s − 0.384·12-s + (0.971 + 2.99i)13-s + (−0.496 − 0.361i)14-s + (−0.952 + 2.93i)16-s + (−0.775 + 2.38i)17-s + (−1.02 + 0.747i)18-s + (3.00 + 2.18i)19-s + ⋯
L(s)  = 1  + (0.277 + 0.854i)2-s + (−0.467 − 0.339i)3-s + (0.155 − 0.113i)4-s + (0.160 − 0.493i)6-s + (−0.147 + 0.107i)7-s + (0.866 + 0.629i)8-s + (0.103 + 0.317i)9-s + (−0.296 − 0.955i)11-s − 0.111·12-s + (0.269 + 0.829i)13-s + (−0.132 − 0.0964i)14-s + (−0.238 + 0.732i)16-s + (−0.188 + 0.578i)17-s + (−0.242 + 0.176i)18-s + (0.690 + 0.501i)19-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((0.634−0.772i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1/2) L(s)=((0.634−0.772i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: 0.634−0.772i0.634 - 0.772i
Analytic conductor: 6.587656.58765
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ825(676,⋅)\chi_{825} (676, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :1/2), 0.634−0.772i)(2,\ 825,\ (\ :1/2),\ 0.634 - 0.772i)

Particular Values

L(1)L(1) ≈\approx 1.64796+0.779059i1.64796 + 0.779059i
L(12)L(\frac12) ≈\approx 1.64796+0.779059i1.64796 + 0.779059i
L(32)L(\frac{3}{2}) 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)
bad3 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
5 1 1
11 1+(0.982+3.16i)T 1 + (0.982 + 3.16i)T
good2 1+(−0.392−1.20i)T+(−1.61+1.17i)T2 1 + (-0.392 - 1.20i)T + (-1.61 + 1.17i)T^{2}
7 1+(0.390−0.284i)T+(2.16−6.65i)T2 1 + (0.390 - 0.284i)T + (2.16 - 6.65i)T^{2}
13 1+(−0.971−2.99i)T+(−10.5+7.64i)T2 1 + (-0.971 - 2.99i)T + (-10.5 + 7.64i)T^{2}
17 1+(0.775−2.38i)T+(−13.7−9.99i)T2 1 + (0.775 - 2.38i)T + (-13.7 - 9.99i)T^{2}
19 1+(−3.00−2.18i)T+(5.87+18.0i)T2 1 + (-3.00 - 2.18i)T + (5.87 + 18.0i)T^{2}
23 1−8.53T+23T2 1 - 8.53T + 23T^{2}
29 1+(−8.07+5.86i)T+(8.96−27.5i)T2 1 + (-8.07 + 5.86i)T + (8.96 - 27.5i)T^{2}
31 1+(1.78+5.50i)T+(−25.0+18.2i)T2 1 + (1.78 + 5.50i)T + (-25.0 + 18.2i)T^{2}
37 1+(2.59−1.88i)T+(11.4−35.1i)T2 1 + (2.59 - 1.88i)T + (11.4 - 35.1i)T^{2}
41 1+(−6.36−4.62i)T+(12.6+38.9i)T2 1 + (-6.36 - 4.62i)T + (12.6 + 38.9i)T^{2}
43 1+5.97T+43T2 1 + 5.97T + 43T^{2}
47 1+(−4.96−3.60i)T+(14.5+44.6i)T2 1 + (-4.96 - 3.60i)T + (14.5 + 44.6i)T^{2}
53 1+(−0.488−1.50i)T+(−42.8+31.1i)T2 1 + (-0.488 - 1.50i)T + (-42.8 + 31.1i)T^{2}
59 1+(−0.305+0.222i)T+(18.2−56.1i)T2 1 + (-0.305 + 0.222i)T + (18.2 - 56.1i)T^{2}
61 1+(−0.929+2.86i)T+(−49.3−35.8i)T2 1 + (-0.929 + 2.86i)T + (-49.3 - 35.8i)T^{2}
67 1+5.98T+67T2 1 + 5.98T + 67T^{2}
71 1+(−2.57+7.93i)T+(−57.4−41.7i)T2 1 + (-2.57 + 7.93i)T + (-57.4 - 41.7i)T^{2}
73 1+(−0.122+0.0889i)T+(22.5−69.4i)T2 1 + (-0.122 + 0.0889i)T + (22.5 - 69.4i)T^{2}
79 1+(−2.31−7.11i)T+(−63.9+46.4i)T2 1 + (-2.31 - 7.11i)T + (-63.9 + 46.4i)T^{2}
83 1+(−1.25+3.85i)T+(−67.1−48.7i)T2 1 + (-1.25 + 3.85i)T + (-67.1 - 48.7i)T^{2}
89 1+17.0T+89T2 1 + 17.0T + 89T^{2}
97 1+(1.11+3.43i)T+(−78.4+57.0i)T2 1 + (1.11 + 3.43i)T + (-78.4 + 57.0i)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

−10.55584700088945633922537154584, −9.409866880232040943785886277427, −8.350921829980655596859205860686, −7.63933405316790799692913560234, −6.63444286825327119916771930781, −6.13105492262702106198892871570, −5.31028214019973325726287037660, −4.31383592529091578872082483655, −2.77941937835163091781031311239, −1.27141286439250328579353724594, 1.08955816111322313245081328138, 2.67149825737576848838643037653, 3.45223523544761934921617866136, 4.71368735898962627655235515505, 5.29846654781782780417272289231, 6.93809830552465897861289442143, 7.16958655340606026817340854501, 8.582799008407923556793125126682, 9.578404965181297101957988536632, 10.45081058651424266528133952312

Graph of the ZZ-function along the critical line