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Properties

Label 2-666-111.2-c1-0-10
Degree 22
Conductor 666666
Sign 0.994+0.106i0.994 + 0.106i
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 + (0.700 + 0.326i)5-s + (4.53 − 1.65i)7-s + (0.258 − 0.965i)8-s + (−0.386 + 0.668i)10-s + (−2.42 − 4.19i)11-s + (−2.50 − 3.57i)13-s + (1.25 + 4.66i)14-s + (0.939 + 0.342i)16-s + (1.37 − 1.96i)17-s + (5.73 − 0.501i)19-s + (−0.632 − 0.443i)20-s + (4.39 − 2.04i)22-s + (−5.80 + 1.55i)23-s + ⋯
L(s)  = 1  + (−0.0616 + 0.704i)2-s + (−0.492 − 0.0868i)4-s + (0.313 + 0.145i)5-s + (1.71 − 0.624i)7-s + (0.0915 − 0.341i)8-s + (−0.122 + 0.211i)10-s + (−0.730 − 1.26i)11-s + (−0.694 − 0.992i)13-s + (0.334 + 1.24i)14-s + (0.234 + 0.0855i)16-s + (0.334 − 0.477i)17-s + (1.31 − 0.115i)19-s + (−0.141 − 0.0990i)20-s + (0.936 − 0.436i)22-s + (−1.21 + 0.324i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.994+0.106i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.994+0.106i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.994+0.106i0.994 + 0.106i
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), 0.994+0.106i)(2,\ 666,\ (\ :1/2),\ 0.994 + 0.106i)

Particular Values

L(1)L(1) ≈\approx 1.59032−0.0853091i1.59032 - 0.0853091i
L(12)L(\frac12) ≈\approx 1.59032−0.0853091i1.59032 - 0.0853091i
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+(−2.35−5.60i)T 1 + (-2.35 - 5.60i)T
good5 1+(−0.700−0.326i)T+(3.21+3.83i)T2 1 + (-0.700 - 0.326i)T + (3.21 + 3.83i)T^{2}
7 1+(−4.53+1.65i)T+(5.36−4.49i)T2 1 + (-4.53 + 1.65i)T + (5.36 - 4.49i)T^{2}
11 1+(2.42+4.19i)T+(−5.5+9.52i)T2 1 + (2.42 + 4.19i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.50+3.57i)T+(−4.44+12.2i)T2 1 + (2.50 + 3.57i)T + (-4.44 + 12.2i)T^{2}
17 1+(−1.37+1.96i)T+(−5.81−15.9i)T2 1 + (-1.37 + 1.96i)T + (-5.81 - 15.9i)T^{2}
19 1+(−5.73+0.501i)T+(18.7−3.29i)T2 1 + (-5.73 + 0.501i)T + (18.7 - 3.29i)T^{2}
23 1+(5.80−1.55i)T+(19.9−11.5i)T2 1 + (5.80 - 1.55i)T + (19.9 - 11.5i)T^{2}
29 1+(−1.94−0.520i)T+(25.1+14.5i)T2 1 + (-1.94 - 0.520i)T + (25.1 + 14.5i)T^{2}
31 1+(2.21+2.21i)T+31iT2 1 + (2.21 + 2.21i)T + 31iT^{2}
41 1+(1.55−8.84i)T+(−38.5−14.0i)T2 1 + (1.55 - 8.84i)T + (-38.5 - 14.0i)T^{2}
43 1+(−5.76+5.76i)T−43iT2 1 + (-5.76 + 5.76i)T - 43iT^{2}
47 1+(−8.09−4.67i)T+(23.5+40.7i)T2 1 + (-8.09 - 4.67i)T + (23.5 + 40.7i)T^{2}
53 1+(−0.878+2.41i)T+(−40.6−34.0i)T2 1 + (-0.878 + 2.41i)T + (-40.6 - 34.0i)T^{2}
59 1+(−3.50−7.51i)T+(−37.9+45.1i)T2 1 + (-3.50 - 7.51i)T + (-37.9 + 45.1i)T^{2}
61 1+(3.07−2.15i)T+(20.8−57.3i)T2 1 + (3.07 - 2.15i)T + (20.8 - 57.3i)T^{2}
67 1+(−4.27−11.7i)T+(−51.3+43.0i)T2 1 + (-4.27 - 11.7i)T + (-51.3 + 43.0i)T^{2}
71 1+(−6.82+8.13i)T+(−12.3−69.9i)T2 1 + (-6.82 + 8.13i)T + (-12.3 - 69.9i)T^{2}
73 1−5.96iT−73T2 1 - 5.96iT - 73T^{2}
79 1+(5.34−11.4i)T+(−50.7−60.5i)T2 1 + (5.34 - 11.4i)T + (-50.7 - 60.5i)T^{2}
83 1+(7.54−1.32i)T+(77.9−28.3i)T2 1 + (7.54 - 1.32i)T + (77.9 - 28.3i)T^{2}
89 1+(8.54−3.98i)T+(57.2−68.1i)T2 1 + (8.54 - 3.98i)T + (57.2 - 68.1i)T^{2}
97 1+(0.616+2.30i)T+(−84.0+48.5i)T2 1 + (0.616 + 2.30i)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

−10.39963147887729987211691124458, −9.718589316742069574656643484481, −8.318543722569496369108722445211, −7.935876557063147740694375924402, −7.25472978652460767740899158241, −5.71335392981183234237030337527, −5.33730080082233370093608828257, −4.20715632134125164433991710545, −2.74159571757286144782158211099, −0.951860138645242554113452201642, 1.73482657963743804581310465442, 2.28684806745712378893729677947, 4.09928220529845719730496240419, 4.97855838042070299254215779789, 5.62664756695663617447180166607, 7.37981354197648554725433295759, 7.906115259097675550756245607921, 8.999970681750513351734863541067, 9.712350186179226493230755177292, 10.54738479038107834215712968272

Graph of the ZZ-function along the critical line