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Properties

Label 2-931-133.68-c0-0-0
Degree 22
Conductor 931931
Sign 0.962−0.272i0.962 - 0.272i
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)2-s + i·3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s − 8-s + (0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)16-s + i·17-s + (0.866 + 0.5i)19-s + 23-s − i·24-s + (−0.866 − 0.499i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + i·3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s − 8-s + (0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)16-s + i·17-s + (0.866 + 0.5i)19-s + 23-s − i·24-s + (−0.866 − 0.499i)26-s + i·27-s + ⋯

Functional equation

Λ(s)=(931s/2 ΓC(s) L(s)=((0.962−0.272i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(931s/2 ΓC(s) L(s)=((0.962−0.272i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 931931    =    72⋅197^{2} \cdot 19
Sign: 0.962−0.272i0.962 - 0.272i
Analytic conductor: 0.4646290.464629
Root analytic conductor: 0.6816370.681637
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ931(68,⋅)\chi_{931} (68, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 931, ( :0), 0.962−0.272i)(2,\ 931,\ (\ :0),\ 0.962 - 0.272i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.71357663120.7135766312
L(12)L(\frac12) ≈\approx 0.71357663120.7135766312
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
good2 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
3 1−iT−T2 1 - iT - T^{2}
5 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
17 1−iT−T2 1 - iT - T^{2}
23 1−T+T2 1 - T + T^{2}
29 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)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+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
43 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1−iT−T2 1 - iT - T^{2}
53 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
59 1+iT−T2 1 + iT - T^{2}
61 1+iT−T2 1 + iT - T^{2}
67 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
73 1+iT−T2 1 + iT - T^{2}
79 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+iT−T2 1 + iT - T^{2}
97 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)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

−10.43676262681067968055608137782, −9.731789625085636846671381449877, −8.883899026118061759349448557131, −8.087318345926855616247234391069, −6.97387248822256457515986040301, −5.91838539305806821428711560719, −4.83793345792158278743608965280, −3.49483453170177207935044155537, −3.30682720694854151566331497045, −1.46401039924033847463452780961, 0.948575539220244715818027798531, 2.71183074506747621594498340452, 3.97796069901549526153324061350, 5.19908999135408060994142871358, 6.36620652026159446138196942441, 7.08697363604737370016753112194, 7.54383511545847247837709464738, 8.426873909798169770306747984297, 8.922767286655287649361857608899, 9.965323104378623187342984000636

Graph of the ZZ-function along the critical line