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Properties

Label 2-665-19.7-c1-0-26
Degree 22
Conductor 665665
Sign 0.238+0.971i0.238 + 0.971i
Analytic cond. 5.310055.31005
Root an. cond. 2.304352.30435
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  + (−0.675 − 1.17i)3-s + (1 − 1.73i)4-s + (0.5 + 0.866i)5-s + 7-s + (0.586 − 1.01i)9-s + 5.17·11-s − 2.70·12-s + (−1 + 1.73i)13-s + (0.675 − 1.17i)15-s + (−1.99 − 3.46i)16-s + (−0.351 − 0.609i)17-s + (1.91 + 3.91i)19-s + 1.99·20-s + (−0.675 − 1.17i)21-s + (2.76 − 4.78i)23-s + ⋯
L(s)  = 1  + (−0.390 − 0.675i)3-s + (0.5 − 0.866i)4-s + (0.223 + 0.387i)5-s + 0.377·7-s + (0.195 − 0.338i)9-s + 1.55·11-s − 0.780·12-s + (−0.277 + 0.480i)13-s + (0.174 − 0.302i)15-s + (−0.499 − 0.866i)16-s + (−0.0853 − 0.147i)17-s + (0.438 + 0.898i)19-s + 0.447·20-s + (−0.147 − 0.255i)21-s + (0.575 − 0.997i)23-s + ⋯

Functional equation

Λ(s)=(665s/2 ΓC(s) L(s)=((0.238+0.971i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(665s/2 ΓC(s+1/2) L(s)=((0.238+0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 665665    =    5⋅7⋅195 \cdot 7 \cdot 19
Sign: 0.238+0.971i0.238 + 0.971i
Analytic conductor: 5.310055.31005
Root analytic conductor: 2.304352.30435
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ665(596,⋅)\chi_{665} (596, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 665, ( :1/2), 0.238+0.971i)(2,\ 665,\ (\ :1/2),\ 0.238 + 0.971i)

Particular Values

L(1)L(1) ≈\approx 1.35581−1.06319i1.35581 - 1.06319i
L(12)L(\frac12) ≈\approx 1.35581−1.06319i1.35581 - 1.06319i
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)
bad5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1−T 1 - T
19 1+(−1.91−3.91i)T 1 + (-1.91 - 3.91i)T
good2 1+(−1+1.73i)T2 1 + (-1 + 1.73i)T^{2}
3 1+(0.675+1.17i)T+(−1.5+2.59i)T2 1 + (0.675 + 1.17i)T + (-1.5 + 2.59i)T^{2}
11 1−5.17T+11T2 1 - 5.17T + 11T^{2}
13 1+(1−1.73i)T+(−6.5−11.2i)T2 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2}
17 1+(0.351+0.609i)T+(−8.5+14.7i)T2 1 + (0.351 + 0.609i)T + (-8.5 + 14.7i)T^{2}
23 1+(−2.76+4.78i)T+(−11.5−19.9i)T2 1 + (-2.76 + 4.78i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.85−3.20i)T+(−14.5−25.1i)T2 1 + (1.85 - 3.20i)T + (-14.5 - 25.1i)T^{2}
31 1−3.82T+31T2 1 - 3.82T + 31T^{2}
37 1+2.82T+37T2 1 + 2.82T + 37T^{2}
41 1+(4.08+7.07i)T+(−20.5+35.5i)T2 1 + (4.08 + 7.07i)T + (-20.5 + 35.5i)T^{2}
43 1+(4+6.92i)T+(−21.5+37.2i)T2 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.49−6.05i)T+(−23.5−40.7i)T2 1 + (3.49 - 6.05i)T + (-23.5 - 40.7i)T^{2}
53 1+(−2.41+4.17i)T+(−26.5−45.8i)T2 1 + (-2.41 + 4.17i)T + (-26.5 - 45.8i)T^{2}
59 1+(2.82+4.89i)T+(−29.5+51.0i)T2 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2.52+4.37i)T+(−30.5−52.8i)T2 1 + (-2.52 + 4.37i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.82−10.0i)T+(−33.5−58.0i)T2 1 + (5.82 - 10.0i)T + (-33.5 - 58.0i)T^{2}
71 1+(−0.413−0.716i)T+(−35.5+61.4i)T2 1 + (-0.413 - 0.716i)T + (-35.5 + 61.4i)T^{2}
73 1+(−5.84−10.1i)T+(−36.5+63.2i)T2 1 + (-5.84 - 10.1i)T + (-36.5 + 63.2i)T^{2}
79 1+(−1.93−3.35i)T+(−39.5+68.4i)T2 1 + (-1.93 - 3.35i)T + (-39.5 + 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(−4.26+7.38i)T+(−44.5−77.0i)T2 1 + (-4.26 + 7.38i)T + (-44.5 - 77.0i)T^{2}
97 1+(−4.87−8.44i)T+(−48.5+84.0i)T2 1 + (-4.87 - 8.44i)T + (-48.5 + 84.0i)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

−10.35737301052584921855618713844, −9.598524599242189191034964555744, −8.729091166602708194184200045242, −7.25132223526524975865937485074, −6.72833433249717229857829809153, −6.12217054302690278921154839684, −5.04132452468578451201922401357, −3.70281177899608783973360560580, −2.02994869980025784886523812867, −1.12174822512070078898066327636, 1.64414928021479718420897213953, 3.18964867804754293255362322908, 4.26807698214335605336871959369, 5.05335202002362261970458141249, 6.25889617449958587570387990645, 7.21682324352628313418196066245, 8.084858272923133752884982285331, 9.071634547941162552709949619402, 9.782489682407259864431691330564, 10.85916950659726165772674206196

Graph of the ZZ-function along the critical line