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Properties

Label 2-960-960.629-c0-0-0
Degree 22
Conductor 960960
Sign 0.881−0.471i0.881 - 0.471i
Analytic cond. 0.4791020.479102
Root an. cond. 0.6921720.692172
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  + (−0.831 − 0.555i)2-s + (−0.980 + 0.195i)3-s + (0.382 + 0.923i)4-s + (0.555 + 0.831i)5-s + (0.923 + 0.382i)6-s + (0.195 − 0.980i)8-s + (0.923 − 0.382i)9-s − i·10-s + (−0.555 − 0.831i)12-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)16-s + (0.785 − 0.785i)17-s + (−0.980 − 0.195i)18-s + (−0.324 − 0.216i)19-s + (−0.555 + 0.831i)20-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)2-s + (−0.980 + 0.195i)3-s + (0.382 + 0.923i)4-s + (0.555 + 0.831i)5-s + (0.923 + 0.382i)6-s + (0.195 − 0.980i)8-s + (0.923 − 0.382i)9-s − i·10-s + (−0.555 − 0.831i)12-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)16-s + (0.785 − 0.785i)17-s + (−0.980 − 0.195i)18-s + (−0.324 − 0.216i)19-s + (−0.555 + 0.831i)20-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.881−0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s) L(s)=((0.881−0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.881−0.471i0.881 - 0.471i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ960(629,⋅)\chi_{960} (629, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :0), 0.881−0.471i)(2,\ 960,\ (\ :0),\ 0.881 - 0.471i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.56330855700.5633085570
L(12)L(\frac12) ≈\approx 0.56330855700.5633085570
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+(0.831+0.555i)T 1 + (0.831 + 0.555i)T
3 1+(0.980−0.195i)T 1 + (0.980 - 0.195i)T
5 1+(−0.555−0.831i)T 1 + (-0.555 - 0.831i)T
good7 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
11 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
13 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
17 1+(−0.785+0.785i)T−iT2 1 + (-0.785 + 0.785i)T - iT^{2}
19 1+(0.324+0.216i)T+(0.382+0.923i)T2 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2}
23 1+(−0.636−1.53i)T+(−0.707+0.707i)T2 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2}
29 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
31 1−0.765iT−T2 1 - 0.765iT - T^{2}
37 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
41 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
43 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
47 1+(−1.38+1.38i)T−iT2 1 + (-1.38 + 1.38i)T - iT^{2}
53 1+(0.360−1.81i)T+(−0.923−0.382i)T2 1 + (0.360 - 1.81i)T + (-0.923 - 0.382i)T^{2}
59 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
61 1+(0.382−0.0761i)T+(0.923−0.382i)T2 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2}
67 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
71 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
73 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
79 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
83 1+(−1.53−1.02i)T+(0.382+0.923i)T2 1 + (-1.53 - 1.02i)T + (0.382 + 0.923i)T^{2}
89 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)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

−10.44962847862682044303771310985, −9.601705524758622060245368740273, −9.044713987991844106858042146294, −7.52766514209565026159078558523, −7.11482739472531021321192228959, −6.11089198829179025433546590814, −5.18799122950868329483236083859, −3.81167825204261864475170281853, −2.78847861580325377275382927568, −1.36507989036505711871277021616, 0.935864577919110723324970379796, 2.13349022541472722125959876556, 4.34460785101279142741306946355, 5.25734134235439586665093601918, 5.98759864803307841254874746924, 6.62914582419739990277336667158, 7.70872541846262633005002070318, 8.466286843253908714713983970327, 9.336223730049960122066925914546, 10.20112166021110365662864065613

Graph of the ZZ-function along the critical line