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Properties

Label 2-666-111.2-c1-0-5
Degree 22
Conductor 666666
Sign 0.945+0.325i0.945 + 0.325i
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  + (0.0871 − 0.996i)2-s + (−0.984 − 0.173i)4-s + (3.66 + 1.70i)5-s + (−0.294 + 0.107i)7-s + (−0.258 + 0.965i)8-s + (2.02 − 3.49i)10-s + (−0.875 − 1.51i)11-s + (1.84 + 2.64i)13-s + (0.0811 + 0.302i)14-s + (0.939 + 0.342i)16-s + (−0.590 + 0.843i)17-s + (5.81 − 0.508i)19-s + (−3.31 − 2.31i)20-s + (−1.58 + 0.740i)22-s + (−2.42 + 0.648i)23-s + ⋯
L(s)  = 1  + (0.0616 − 0.704i)2-s + (−0.492 − 0.0868i)4-s + (1.63 + 0.763i)5-s + (−0.111 + 0.0405i)7-s + (−0.0915 + 0.341i)8-s + (0.638 − 1.10i)10-s + (−0.264 − 0.457i)11-s + (0.513 + 0.732i)13-s + (0.0216 + 0.0809i)14-s + (0.234 + 0.0855i)16-s + (−0.143 + 0.204i)17-s + (1.33 − 0.116i)19-s + (−0.740 − 0.518i)20-s + (−0.338 + 0.157i)22-s + (−0.504 + 0.135i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.945+0.325i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.945+0.325i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.945+0.325i0.945 + 0.325i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(557,⋅)\chi_{666} (557, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), 0.945+0.325i)(2,\ 666,\ (\ :1/2),\ 0.945 + 0.325i)

Particular Values

L(1)L(1) ≈\approx 1.91028−0.319198i1.91028 - 0.319198i
L(12)L(\frac12) ≈\approx 1.91028−0.319198i1.91028 - 0.319198i
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+(−0.0871+0.996i)T 1 + (-0.0871 + 0.996i)T
3 1 1
37 1+(−5.99−1.00i)T 1 + (-5.99 - 1.00i)T
good5 1+(−3.66−1.70i)T+(3.21+3.83i)T2 1 + (-3.66 - 1.70i)T + (3.21 + 3.83i)T^{2}
7 1+(0.294−0.107i)T+(5.36−4.49i)T2 1 + (0.294 - 0.107i)T + (5.36 - 4.49i)T^{2}
11 1+(0.875+1.51i)T+(−5.5+9.52i)T2 1 + (0.875 + 1.51i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1.84−2.64i)T+(−4.44+12.2i)T2 1 + (-1.84 - 2.64i)T + (-4.44 + 12.2i)T^{2}
17 1+(0.590−0.843i)T+(−5.81−15.9i)T2 1 + (0.590 - 0.843i)T + (-5.81 - 15.9i)T^{2}
19 1+(−5.81+0.508i)T+(18.7−3.29i)T2 1 + (-5.81 + 0.508i)T + (18.7 - 3.29i)T^{2}
23 1+(2.42−0.648i)T+(19.9−11.5i)T2 1 + (2.42 - 0.648i)T + (19.9 - 11.5i)T^{2}
29 1+(2.34+0.627i)T+(25.1+14.5i)T2 1 + (2.34 + 0.627i)T + (25.1 + 14.5i)T^{2}
31 1+(−0.925−0.925i)T+31iT2 1 + (-0.925 - 0.925i)T + 31iT^{2}
41 1+(0.370−2.10i)T+(−38.5−14.0i)T2 1 + (0.370 - 2.10i)T + (-38.5 - 14.0i)T^{2}
43 1+(−4.14+4.14i)T−43iT2 1 + (-4.14 + 4.14i)T - 43iT^{2}
47 1+(3.98+2.30i)T+(23.5+40.7i)T2 1 + (3.98 + 2.30i)T + (23.5 + 40.7i)T^{2}
53 1+(−2.72+7.48i)T+(−40.6−34.0i)T2 1 + (-2.72 + 7.48i)T + (-40.6 - 34.0i)T^{2}
59 1+(−0.432−0.927i)T+(−37.9+45.1i)T2 1 + (-0.432 - 0.927i)T + (-37.9 + 45.1i)T^{2}
61 1+(−0.131+0.0918i)T+(20.8−57.3i)T2 1 + (-0.131 + 0.0918i)T + (20.8 - 57.3i)T^{2}
67 1+(5.46+15.0i)T+(−51.3+43.0i)T2 1 + (5.46 + 15.0i)T + (-51.3 + 43.0i)T^{2}
71 1+(−8.74+10.4i)T+(−12.3−69.9i)T2 1 + (-8.74 + 10.4i)T + (-12.3 - 69.9i)T^{2}
73 1+11.4iT−73T2 1 + 11.4iT - 73T^{2}
79 1+(0.448−0.961i)T+(−50.7−60.5i)T2 1 + (0.448 - 0.961i)T + (-50.7 - 60.5i)T^{2}
83 1+(8.03−1.41i)T+(77.9−28.3i)T2 1 + (8.03 - 1.41i)T + (77.9 - 28.3i)T^{2}
89 1+(12.5−5.83i)T+(57.2−68.1i)T2 1 + (12.5 - 5.83i)T + (57.2 - 68.1i)T^{2}
97 1+(−2.88−10.7i)T+(−84.0+48.5i)T2 1 + (-2.88 - 10.7i)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

−10.52861349585138574679972394284, −9.566002937071762653266844124365, −9.279011333650659693398988153736, −7.956447663714018731860695509410, −6.66352434832245651120043701783, −5.95501640801334908573898821047, −5.07864224496056635922513385605, −3.56908696470741018849220506016, −2.57881610943576905509543523093, −1.53181894685241278159644444123, 1.24205680955113296409147287192, 2.73722681809699600113287617813, 4.36444808895101140052488508934, 5.49090326963033227062148009673, 5.79005995583173582093174098035, 6.92616976337739829697409956759, 7.979009846103988372610681730728, 8.863363377302984057856078577566, 9.740724222956699664857860831565, 10.06854168384367342076736374213

Graph of the ZZ-function along the critical line