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Properties

Label 2-666-37.11-c1-0-6
Degree 22
Conductor 666666
Sign 0.367+0.929i0.367 + 0.929i
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.633 − 1.09i)7-s + 0.999i·8-s + 0.999·10-s − 4.73·11-s + (4.73 + 2.73i)13-s + 1.26i·14-s + (−0.5 − 0.866i)16-s + (6.69 − 3.86i)17-s + (−5.83 − 3.36i)19-s + (−0.866 + 0.499i)20-s + (4.09 − 2.36i)22-s − 1.46i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.239 − 0.415i)7-s + 0.353i·8-s + 0.316·10-s − 1.42·11-s + (1.31 + 0.757i)13-s + 0.338i·14-s + (−0.125 − 0.216i)16-s + (1.62 − 0.937i)17-s + (−1.33 − 0.772i)19-s + (−0.193 + 0.111i)20-s + (0.873 − 0.504i)22-s − 0.305i·23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.367+0.929i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.367+0.929i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.367+0.929i0.367 + 0.929i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(307,⋅)\chi_{666} (307, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), 0.367+0.929i)(2,\ 666,\ (\ :1/2),\ 0.367 + 0.929i)

Particular Values

L(1)L(1) ≈\approx 0.687787−0.467668i0.687787 - 0.467668i
L(12)L(\frac12) ≈\approx 0.687787−0.467668i0.687787 - 0.467668i
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)
bad2 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
3 1 1
37 1+(6.06−0.5i)T 1 + (6.06 - 0.5i)T
good5 1+(0.866+0.5i)T+(2.5+4.33i)T2 1 + (0.866 + 0.5i)T + (2.5 + 4.33i)T^{2}
7 1+(−0.633+1.09i)T+(−3.5−6.06i)T2 1 + (-0.633 + 1.09i)T + (-3.5 - 6.06i)T^{2}
11 1+4.73T+11T2 1 + 4.73T + 11T^{2}
13 1+(−4.73−2.73i)T+(6.5+11.2i)T2 1 + (-4.73 - 2.73i)T + (6.5 + 11.2i)T^{2}
17 1+(−6.69+3.86i)T+(8.5−14.7i)T2 1 + (-6.69 + 3.86i)T + (8.5 - 14.7i)T^{2}
19 1+(5.83+3.36i)T+(9.5+16.4i)T2 1 + (5.83 + 3.36i)T + (9.5 + 16.4i)T^{2}
23 1+1.46iT−23T2 1 + 1.46iT - 23T^{2}
29 1−0.464iT−29T2 1 - 0.464iT - 29T^{2}
31 1+6.19iT−31T2 1 + 6.19iT - 31T^{2}
41 1+(−4.23+7.33i)T+(−20.5−35.5i)T2 1 + (-4.23 + 7.33i)T + (-20.5 - 35.5i)T^{2}
43 1+5.26iT−43T2 1 + 5.26iT - 43T^{2}
47 1−5.26T+47T2 1 - 5.26T + 47T^{2}
53 1+(4.46+7.73i)T+(−26.5+45.8i)T2 1 + (4.46 + 7.73i)T + (-26.5 + 45.8i)T^{2}
59 1+(−6+3.46i)T+(29.5−51.0i)T2 1 + (-6 + 3.46i)T + (29.5 - 51.0i)T^{2}
61 1+(7.33+4.23i)T+(30.5+52.8i)T2 1 + (7.33 + 4.23i)T + (30.5 + 52.8i)T^{2}
67 1+(−1.83+3.16i)T+(−33.5−58.0i)T2 1 + (-1.83 + 3.16i)T + (-33.5 - 58.0i)T^{2}
71 1+(2.63−4.56i)T+(−35.5−61.4i)T2 1 + (2.63 - 4.56i)T + (-35.5 - 61.4i)T^{2}
73 1−12.3T+73T2 1 - 12.3T + 73T^{2}
79 1+(−5.36−3.09i)T+(39.5+68.4i)T2 1 + (-5.36 - 3.09i)T + (39.5 + 68.4i)T^{2}
83 1+(3.26+5.66i)T+(−41.5+71.8i)T2 1 + (3.26 + 5.66i)T + (-41.5 + 71.8i)T^{2}
89 1+(7.5−4.33i)T+(44.5−77.0i)T2 1 + (7.5 - 4.33i)T + (44.5 - 77.0i)T^{2}
97 1+1.33iT−97T2 1 + 1.33iT - 97T^{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

−10.44564126795321333798997209030, −9.393481573054499433930520211212, −8.458777884376278679024243775286, −7.86520830918177183790695438092, −7.01680163453890182028361157901, −5.93711951590814295462850842329, −4.95057923473085395294709247875, −3.81357603511856991176926327972, −2.28407665196525760525246953201, −0.57061564323279276290378339986, 1.48086640914669884695660055302, 2.95859953069710348367116392654, 3.82567464463763001301904548577, 5.40325040303042630695565998647, 6.14259278086528115312273468744, 7.62488358279123191196828655305, 8.095658651653836932364881818490, 8.764961462413206188894536689844, 10.13241743961508650058624511859, 10.56283540594214863763072051322

Graph of the ZZ-function along the critical line