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Properties

Label 2-666-111.14-c1-0-7
Degree 22
Conductor 666666
Sign −0.358+0.933i-0.358 + 0.933i
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.653 + 2.43i)5-s + (−1.76 − 3.05i)7-s + (−0.707 − 0.707i)8-s − 2.52·10-s − 2.53·11-s + (−2.84 − 0.762i)13-s + (2.49 − 2.49i)14-s + (0.500 − 0.866i)16-s + (1.73 − 0.463i)17-s + (−3.53 − 0.946i)19-s + (−0.653 − 2.43i)20-s + (−0.656 − 2.44i)22-s + (−3.15 − 3.15i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.292 + 1.09i)5-s + (−0.666 − 1.15i)7-s + (−0.249 − 0.249i)8-s − 0.798·10-s − 0.764·11-s + (−0.788 − 0.211i)13-s + (0.666 − 0.666i)14-s + (0.125 − 0.216i)16-s + (0.419 − 0.112i)17-s + (−0.809 − 0.217i)19-s + (−0.146 − 0.545i)20-s + (−0.139 − 0.521i)22-s + (−0.658 − 0.658i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((−0.358+0.933i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((−0.358+0.933i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: −0.358+0.933i-0.358 + 0.933i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(125,⋅)\chi_{666} (125, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), −0.358+0.933i)(2,\ 666,\ (\ :1/2),\ -0.358 + 0.933i)

Particular Values

L(1)L(1) ≈\approx 0.0542814−0.0789563i0.0542814 - 0.0789563i
L(12)L(\frac12) ≈\approx 0.0542814−0.0789563i0.0542814 - 0.0789563i
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+(−0.258−0.965i)T 1 + (-0.258 - 0.965i)T
3 1 1
37 1+(6.08−0.00578i)T 1 + (6.08 - 0.00578i)T
good5 1+(0.653−2.43i)T+(−4.33−2.5i)T2 1 + (0.653 - 2.43i)T + (-4.33 - 2.5i)T^{2}
7 1+(1.76+3.05i)T+(−3.5+6.06i)T2 1 + (1.76 + 3.05i)T + (-3.5 + 6.06i)T^{2}
11 1+2.53T+11T2 1 + 2.53T + 11T^{2}
13 1+(2.84+0.762i)T+(11.2+6.5i)T2 1 + (2.84 + 0.762i)T + (11.2 + 6.5i)T^{2}
17 1+(−1.73+0.463i)T+(14.7−8.5i)T2 1 + (-1.73 + 0.463i)T + (14.7 - 8.5i)T^{2}
19 1+(3.53+0.946i)T+(16.4+9.5i)T2 1 + (3.53 + 0.946i)T + (16.4 + 9.5i)T^{2}
23 1+(3.15+3.15i)T+23iT2 1 + (3.15 + 3.15i)T + 23iT^{2}
29 1+(0.0393−0.0393i)T−29iT2 1 + (0.0393 - 0.0393i)T - 29iT^{2}
31 1+(3.02+3.02i)T+31iT2 1 + (3.02 + 3.02i)T + 31iT^{2}
41 1+(−4.63−8.02i)T+(−20.5+35.5i)T2 1 + (-4.63 - 8.02i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.234+0.234i)T−43iT2 1 + (-0.234 + 0.234i)T - 43iT^{2}
47 1+4.39iT−47T2 1 + 4.39iT - 47T^{2}
53 1+(−3.09−1.78i)T+(26.5+45.8i)T2 1 + (-3.09 - 1.78i)T + (26.5 + 45.8i)T^{2}
59 1+(9.18−2.46i)T+(51.0−29.5i)T2 1 + (9.18 - 2.46i)T + (51.0 - 29.5i)T^{2}
61 1+(−2.55+9.52i)T+(−52.8−30.5i)T2 1 + (-2.55 + 9.52i)T + (-52.8 - 30.5i)T^{2}
67 1+(13.5−7.80i)T+(33.5−58.0i)T2 1 + (13.5 - 7.80i)T + (33.5 - 58.0i)T^{2}
71 1+(3.56−2.05i)T+(35.5−61.4i)T2 1 + (3.56 - 2.05i)T + (35.5 - 61.4i)T^{2}
73 1+4.09iT−73T2 1 + 4.09iT - 73T^{2}
79 1+(−2.84−0.762i)T+(68.4+39.5i)T2 1 + (-2.84 - 0.762i)T + (68.4 + 39.5i)T^{2}
83 1+(7.86+4.54i)T+(41.5+71.8i)T2 1 + (7.86 + 4.54i)T + (41.5 + 71.8i)T^{2}
89 1+(−3.78−14.1i)T+(−77.0+44.5i)T2 1 + (-3.78 - 14.1i)T + (-77.0 + 44.5i)T^{2}
97 1+(0.841−0.841i)T−97iT2 1 + (0.841 - 0.841i)T - 97iT^{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

−10.38052756657648434473830580809, −9.506276679685842714451361288185, −8.162727912242060737507520843938, −7.39527366704529346084038574123, −6.86995946153389996326926008449, −5.96053706999823549901230430620, −4.68085289895640541727176438744, −3.67465967626637059888483013426, −2.68559317459148088107265259495, −0.04515174853428826234087614038, 1.90115294399461783874900591822, 3.03682928393788141046664146063, 4.30166319953181783098753243742, 5.27114982356675595507044390392, 5.93753181383144319698537450689, 7.43300867247320274712469319711, 8.515080629940045051635058000523, 9.070127722000197504055680085796, 9.896091021939731938422202659763, 10.76129785504004006157738200386

Graph of the ZZ-function along the critical line