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Properties

Label 2-666-111.17-c1-0-6
Degree 22
Conductor 666666
Sign 0.604−0.796i0.604 - 0.796i
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  + (−0.573 + 0.819i)2-s + (−0.342 − 0.939i)4-s + (3.55 − 0.310i)5-s + (3.11 + 2.61i)7-s + (0.965 + 0.258i)8-s + (−1.78 + 3.08i)10-s + (−0.00168 − 0.00292i)11-s + (1.48 + 0.692i)13-s + (−3.92 + 1.05i)14-s + (−0.766 + 0.642i)16-s + (−6.72 + 3.13i)17-s + (5.40 − 3.78i)19-s + (−1.50 − 3.23i)20-s + (0.00335 + 0.000293i)22-s + (−1.87 − 7.01i)23-s + ⋯
L(s)  = 1  + (−0.405 + 0.579i)2-s + (−0.171 − 0.469i)4-s + (1.58 − 0.138i)5-s + (1.17 + 0.987i)7-s + (0.341 + 0.0915i)8-s + (−0.563 + 0.976i)10-s + (−0.000508 − 0.000880i)11-s + (0.411 + 0.192i)13-s + (−1.04 + 0.281i)14-s + (−0.191 + 0.160i)16-s + (−1.62 + 0.760i)17-s + (1.24 − 0.868i)19-s + (−0.336 − 0.722i)20-s + (0.000716 + 6.26e−5i)22-s + (−0.391 − 1.46i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.604−0.796i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.604−0.796i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.604−0.796i0.604 - 0.796i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(17,⋅)\chi_{666} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), 0.604−0.796i)(2,\ 666,\ (\ :1/2),\ 0.604 - 0.796i)

Particular Values

L(1)L(1) ≈\approx 1.57618+0.782060i1.57618 + 0.782060i
L(12)L(\frac12) ≈\approx 1.57618+0.782060i1.57618 + 0.782060i
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+(0.573−0.819i)T 1 + (0.573 - 0.819i)T
3 1 1
37 1+(−5.96+1.17i)T 1 + (-5.96 + 1.17i)T
good5 1+(−3.55+0.310i)T+(4.92−0.868i)T2 1 + (-3.55 + 0.310i)T + (4.92 - 0.868i)T^{2}
7 1+(−3.11−2.61i)T+(1.21+6.89i)T2 1 + (-3.11 - 2.61i)T + (1.21 + 6.89i)T^{2}
11 1+(0.00168+0.00292i)T+(−5.5+9.52i)T2 1 + (0.00168 + 0.00292i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1.48−0.692i)T+(8.35+9.95i)T2 1 + (-1.48 - 0.692i)T + (8.35 + 9.95i)T^{2}
17 1+(6.72−3.13i)T+(10.9−13.0i)T2 1 + (6.72 - 3.13i)T + (10.9 - 13.0i)T^{2}
19 1+(−5.40+3.78i)T+(6.49−17.8i)T2 1 + (-5.40 + 3.78i)T + (6.49 - 17.8i)T^{2}
23 1+(1.87+7.01i)T+(−19.9+11.5i)T2 1 + (1.87 + 7.01i)T + (-19.9 + 11.5i)T^{2}
29 1+(−0.451+1.68i)T+(−25.1−14.5i)T2 1 + (-0.451 + 1.68i)T + (-25.1 - 14.5i)T^{2}
31 1+(6.34−6.34i)T−31iT2 1 + (6.34 - 6.34i)T - 31iT^{2}
41 1+(8.09−2.94i)T+(31.4−26.3i)T2 1 + (8.09 - 2.94i)T + (31.4 - 26.3i)T^{2}
43 1+(3.92+3.92i)T+43iT2 1 + (3.92 + 3.92i)T + 43iT^{2}
47 1+(−0.830−0.479i)T+(23.5+40.7i)T2 1 + (-0.830 - 0.479i)T + (23.5 + 40.7i)T^{2}
53 1+(−7.38−8.80i)T+(−9.20+52.1i)T2 1 + (-7.38 - 8.80i)T + (-9.20 + 52.1i)T^{2}
59 1+(−0.918+10.4i)T+(−58.1−10.2i)T2 1 + (-0.918 + 10.4i)T + (-58.1 - 10.2i)T^{2}
61 1+(4.66−10.0i)T+(−39.2−46.7i)T2 1 + (4.66 - 10.0i)T + (-39.2 - 46.7i)T^{2}
67 1+(−6.84+8.15i)T+(−11.6−65.9i)T2 1 + (-6.84 + 8.15i)T + (-11.6 - 65.9i)T^{2}
71 1+(6.56+1.15i)T+(66.7+24.2i)T2 1 + (6.56 + 1.15i)T + (66.7 + 24.2i)T^{2}
73 1−1.04iT−73T2 1 - 1.04iT - 73T^{2}
79 1+(−0.482−5.51i)T+(−77.7+13.7i)T2 1 + (-0.482 - 5.51i)T + (-77.7 + 13.7i)T^{2}
83 1+(0.623−1.71i)T+(−63.5−53.3i)T2 1 + (0.623 - 1.71i)T + (-63.5 - 53.3i)T^{2}
89 1+(−3.68−0.322i)T+(87.6+15.4i)T2 1 + (-3.68 - 0.322i)T + (87.6 + 15.4i)T^{2}
97 1+(4.41−1.18i)T+(84.0−48.5i)T2 1 + (4.41 - 1.18i)T + (84.0 - 48.5i)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

−10.57418565574738507736876536882, −9.523883991506982607859041747288, −8.812557352648763362421677493967, −8.393105638659608257067840246812, −6.95451118930837819917732903005, −6.14795279714029261345233328330, −5.36463154055608813778803278783, −4.61369635983886884646666836768, −2.43528158632527716350426814694, −1.60339078156321831662609827736, 1.33119681068941289755539490713, 2.16025199402190346911856046325, 3.65331563545021361257802836970, 4.89671110800636781151159922479, 5.77457167000576629906680353412, 7.02179621023483124388654621295, 7.81397691307231715227903780406, 8.905857876144214569354396725861, 9.706440157704821836141107902373, 10.28213042592635947888343937092

Graph of the ZZ-function along the critical line