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Properties

Label 2-931-133.37-c0-0-0
Degree 22
Conductor 931931
Sign 0.605−0.795i0.605 - 0.795i
Analytic cond. 0.4646290.464629
Root an. cond. 0.6816370.681637
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  + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s + (0.5 − 0.866i)19-s + 0.999·20-s + (0.5 − 0.866i)23-s + 0.999·36-s − 43-s + (0.499 − 0.866i)44-s + (−0.499 − 0.866i)45-s + (−0.5 + 0.866i)47-s − 0.999·55-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s + (0.5 − 0.866i)19-s + 0.999·20-s + (0.5 − 0.866i)23-s + 0.999·36-s − 43-s + (0.499 − 0.866i)44-s + (−0.499 − 0.866i)45-s + (−0.5 + 0.866i)47-s − 0.999·55-s + ⋯

Functional equation

Λ(s)=(931s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(931s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 931931    =    72⋅197^{2} \cdot 19
Sign: 0.605−0.795i0.605 - 0.795i
Analytic conductor: 0.4646290.464629
Root analytic conductor: 0.6816370.681637
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ931(569,⋅)\chi_{931} (569, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 931, ( :0), 0.605−0.795i)(2,\ 931,\ (\ :0),\ 0.605 - 0.795i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.78167872390.7816787239
L(12)L(\frac12) ≈\approx 0.78167872390.7816787239
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)
bad7 1 1
19 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good2 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
3 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
5 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
23 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1−T+T2 1 - T + T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1−T2 1 - 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

−10.51965155021030884811589356441, −9.696833605043405474458211810637, −8.737757858825089411608211277743, −7.88583062254236102062964680503, −6.94914445839036822933054455115, −6.10500281671357042875532660945, −5.11880320498645872229286258147, −4.24823142207975782074527625819, −3.04881507901113434033220382517, −1.66431586233228758296887567353, 0.846219027675123186127661269350, 3.16011202294472921603183823811, 3.62655782214096131430972004391, 4.83871484909189759861215890156, 5.62429856210009566655797598174, 6.90565084269467785278640599694, 7.83315833468756567680668419051, 8.494899466022207883699125956180, 9.212757235108250744678764577279, 9.772764656172235479344526227283

Graph of the ZZ-function along the critical line