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Properties

Label 2-644-644.83-c0-0-1
Degree 22
Conductor 644644
Sign 0.264+0.964i0.264 + 0.964i
Analytic cond. 0.3213970.321397
Root an. cond. 0.5669190.566919
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  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (0.142 − 0.989i)9-s + (−1.25 − 0.368i)11-s + (−0.654 + 0.755i)14-s + (0.415 + 0.909i)16-s + (−0.415 + 0.909i)18-s + (1.10 + 0.708i)22-s + (−0.142 − 0.989i)23-s + (0.959 − 0.281i)25-s + (0.841 − 0.540i)28-s + (1.61 + 1.03i)29-s + (−0.142 − 0.989i)32-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (0.142 − 0.989i)9-s + (−1.25 − 0.368i)11-s + (−0.654 + 0.755i)14-s + (0.415 + 0.909i)16-s + (−0.415 + 0.909i)18-s + (1.10 + 0.708i)22-s + (−0.142 − 0.989i)23-s + (0.959 − 0.281i)25-s + (0.841 − 0.540i)28-s + (1.61 + 1.03i)29-s + (−0.142 − 0.989i)32-s + ⋯

Functional equation

Λ(s)=(644s/2 ΓC(s) L(s)=((0.264+0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(644s/2 ΓC(s) L(s)=((0.264+0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 644644    =    22⋅7⋅232^{2} \cdot 7 \cdot 23
Sign: 0.264+0.964i0.264 + 0.964i
Analytic conductor: 0.3213970.321397
Root analytic conductor: 0.5669190.566919
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ644(83,⋅)\chi_{644} (83, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 644, ( :0), 0.264+0.964i)(2,\ 644,\ (\ :0),\ 0.264 + 0.964i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.59421164490.5942116449
L(12)L(\frac12) ≈\approx 0.59421164490.5942116449
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+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
7 1+(−0.415+0.909i)T 1 + (-0.415 + 0.909i)T
23 1+(0.142+0.989i)T 1 + (0.142 + 0.989i)T
good3 1+(−0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
5 1+(−0.959+0.281i)T2 1 + (-0.959 + 0.281i)T^{2}
11 1+(1.25+0.368i)T+(0.841+0.540i)T2 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2}
13 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
17 1+(0.415+0.909i)T2 1 + (0.415 + 0.909i)T^{2}
19 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
29 1+(−1.61−1.03i)T+(0.415+0.909i)T2 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2}
31 1+(−0.142−0.989i)T2 1 + (-0.142 - 0.989i)T^{2}
37 1+(1.80+0.258i)T+(0.959+0.281i)T2 1 + (1.80 + 0.258i)T + (0.959 + 0.281i)T^{2}
41 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
43 1+(−0.186−0.215i)T+(−0.142+0.989i)T2 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−0.512−0.234i)T+(0.654+0.755i)T2 1 + (-0.512 - 0.234i)T + (0.654 + 0.755i)T^{2}
59 1+(−0.654+0.755i)T2 1 + (-0.654 + 0.755i)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+(−0.425−1.45i)T+(−0.841+0.540i)T2 1 + (-0.425 - 1.45i)T + (-0.841 + 0.540i)T^{2}
73 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
79 1+(−0.797−1.74i)T+(−0.654+0.755i)T2 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2}
83 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
89 1+(−0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
97 1+(−0.959+0.281i)T2 1 + (-0.959 + 0.281i)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

−10.52696370836992454374414825984, −9.932354118120941966285645573318, −8.707709211742382346927890115070, −8.230920724596450959705333601923, −7.11039432300326597836829137168, −6.54168099355989227207048677182, −5.06748611798613211944697357102, −3.73961974253959448435030345153, −2.65817140571111455264899115210, −0.952837224183235232463082720439, 1.89364030733974007384833458832, 2.81111660532146483926635998191, 4.95238390413649211298458676733, 5.46716175919257321744828102898, 6.71276129354884281382858015689, 7.75064148593941228571584239836, 8.231461328071374451410396312424, 9.119497221718384241609565698306, 10.17332689025626757073298149365, 10.65829103679777185514566510482

Graph of the ZZ-function along the critical line