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Properties

Label 2-712-712.299-c0-0-0
Degree 22
Conductor 712712
Sign 0.622−0.782i0.622 - 0.782i
Analytic cond. 0.3553340.355334
Root an. cond. 0.5960990.596099
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.142 + 0.989i)2-s + (1.25 − 0.368i)3-s + (−0.959 − 0.281i)4-s + (0.186 + 1.29i)6-s + (0.415 − 0.909i)8-s + (0.601 − 0.386i)9-s + (0.345 + 0.755i)11-s − 1.30·12-s + (0.841 + 0.540i)16-s + (0.0405 + 0.281i)17-s + (0.297 + 0.650i)18-s + (0.698 − 0.449i)19-s + (−0.797 + 0.234i)22-s + (0.186 − 1.29i)24-s + (−0.654 − 0.755i)25-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (1.25 − 0.368i)3-s + (−0.959 − 0.281i)4-s + (0.186 + 1.29i)6-s + (0.415 − 0.909i)8-s + (0.601 − 0.386i)9-s + (0.345 + 0.755i)11-s − 1.30·12-s + (0.841 + 0.540i)16-s + (0.0405 + 0.281i)17-s + (0.297 + 0.650i)18-s + (0.698 − 0.449i)19-s + (−0.797 + 0.234i)22-s + (0.186 − 1.29i)24-s + (−0.654 − 0.755i)25-s + ⋯

Functional equation

Λ(s)=(712s/2 ΓC(s) L(s)=((0.622−0.782i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(712s/2 ΓC(s) L(s)=((0.622−0.782i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 712712    =    23⋅892^{3} \cdot 89
Sign: 0.622−0.782i0.622 - 0.782i
Analytic conductor: 0.3553340.355334
Root analytic conductor: 0.5960990.596099
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ712(299,⋅)\chi_{712} (299, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 712, ( :0), 0.622−0.782i)(2,\ 712,\ (\ :0),\ 0.622 - 0.782i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1900106461.190010646
L(12)L(\frac12) ≈\approx 1.1900106461.190010646
L(1)L(1) 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+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
89 1+(0.654−0.755i)T 1 + (0.654 - 0.755i)T
good3 1+(−1.25+0.368i)T+(0.841−0.540i)T2 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2}
5 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
7 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
11 1+(−0.345−0.755i)T+(−0.654+0.755i)T2 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2}
13 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
17 1+(−0.0405−0.281i)T+(−0.959+0.281i)T2 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2}
19 1+(−0.698+0.449i)T+(0.415−0.909i)T2 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2}
23 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
29 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
31 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
37 1−T2 1 - T^{2}
41 1+(1.91+0.563i)T+(0.841+0.540i)T2 1 + (1.91 + 0.563i)T + (0.841 + 0.540i)T^{2}
43 1+(0.118+0.258i)T+(−0.654+0.755i)T2 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2}
47 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
53 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
59 1+(−0.273−0.0801i)T+(0.841+0.540i)T2 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2}
61 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
67 1+(1.61−0.474i)T+(0.841−0.540i)T2 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2}
71 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
73 1+(1.10+0.708i)T+(0.415+0.909i)T2 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2}
79 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
83 1+(−0.273−1.89i)T+(−0.959+0.281i)T2 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2}
97 1+(−0.698+1.53i)T+(−0.654−0.755i)T2 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)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

−10.28897978433208613963442476255, −9.598083749700947419735927157955, −8.799998435442587017513476209521, −8.162834018412822838870528886126, −7.34967866036938906629110355826, −6.69307085510735417624787415857, −5.47372403399733601554554735782, −4.35204473562697950408046640380, −3.29016378712050823822261355823, −1.80711825186729283861718714979, 1.67365923784166771188416425982, 3.05212854071229786676559595183, 3.52056049804484945678933927214, 4.64861233789756483920775232919, 5.86530410198500265734434016369, 7.46824394133979216320472151609, 8.276232896255895890597842077080, 8.947200210867108794053825681606, 9.629697591498970878286645030122, 10.29829982623371478640165043711

Graph of the ZZ-function along the critical line