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Properties

Label 2-2496-2496.77-c0-0-0
Degree 22
Conductor 24962496
Sign −0.0980−0.995i-0.0980 - 0.995i
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
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  + (−0.471 − 0.881i)2-s + (−0.831 + 0.555i)3-s + (−0.555 + 0.831i)4-s + (−1.72 + 0.344i)5-s + (0.881 + 0.471i)6-s + (0.995 + 0.0980i)8-s + (0.382 − 0.923i)9-s + (1.11 + 1.36i)10-s + (−0.108 + 0.162i)11-s − i·12-s + (0.980 + 0.195i)13-s + (1.24 − 1.24i)15-s + (−0.382 − 0.923i)16-s + (−0.995 + 0.0980i)18-s + (0.674 − 1.62i)20-s + ⋯
L(s)  = 1  + (−0.471 − 0.881i)2-s + (−0.831 + 0.555i)3-s + (−0.555 + 0.831i)4-s + (−1.72 + 0.344i)5-s + (0.881 + 0.471i)6-s + (0.995 + 0.0980i)8-s + (0.382 − 0.923i)9-s + (1.11 + 1.36i)10-s + (−0.108 + 0.162i)11-s − i·12-s + (0.980 + 0.195i)13-s + (1.24 − 1.24i)15-s + (−0.382 − 0.923i)16-s + (−0.995 + 0.0980i)18-s + (0.674 − 1.62i)20-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=((−0.0980−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s) L(s)=((−0.0980−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: −0.0980−0.995i-0.0980 - 0.995i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2496(77,⋅)\chi_{2496} (77, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2496, ( :0), −0.0980−0.995i)(2,\ 2496,\ (\ :0),\ -0.0980 - 0.995i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.26590919810.2659091981
L(12)L(\frac12) ≈\approx 0.26590919810.2659091981
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+(0.471+0.881i)T 1 + (0.471 + 0.881i)T
3 1+(0.831−0.555i)T 1 + (0.831 - 0.555i)T
13 1+(−0.980−0.195i)T 1 + (-0.980 - 0.195i)T
good5 1+(1.72−0.344i)T+(0.923−0.382i)T2 1 + (1.72 - 0.344i)T + (0.923 - 0.382i)T^{2}
7 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
11 1+(0.108−0.162i)T+(−0.382−0.923i)T2 1 + (0.108 - 0.162i)T + (-0.382 - 0.923i)T^{2}
17 1−iT2 1 - iT^{2}
19 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
23 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
29 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
31 1+T2 1 + T^{2}
37 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
41 1+(−0.536−0.222i)T+(0.707+0.707i)T2 1 + (-0.536 - 0.222i)T + (0.707 + 0.707i)T^{2}
43 1+(−0.324−0.216i)T+(0.382+0.923i)T2 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2}
47 1+(1.09+1.09i)T+iT2 1 + (1.09 + 1.09i)T + iT^{2}
53 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
59 1+(0.924−0.183i)T+(0.923−0.382i)T2 1 + (0.924 - 0.183i)T + (0.923 - 0.382i)T^{2}
61 1+(1.53−1.02i)T+(0.382−0.923i)T2 1 + (1.53 - 1.02i)T + (0.382 - 0.923i)T^{2}
67 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
71 1+(−0.761−1.83i)T+(−0.707+0.707i)T2 1 + (-0.761 - 1.83i)T + (-0.707 + 0.707i)T^{2}
73 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
79 1+(0.275−0.275i)T−iT2 1 + (0.275 - 0.275i)T - iT^{2}
83 1+(0.373−1.87i)T+(−0.923−0.382i)T2 1 + (0.373 - 1.87i)T + (-0.923 - 0.382i)T^{2}
89 1+(−1.42+0.591i)T+(0.707−0.707i)T2 1 + (-1.42 + 0.591i)T + (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

−9.392306455193316000331527677319, −8.617770860050407655606497469873, −7.922583976881256315588940589668, −7.18229587729467657802515282489, −6.33668040844271933766059262995, −5.05364005716704274167607296360, −4.22424286204676531371942055114, −3.76009835798916428817294301529, −2.92181283095868249620818154914, −1.15258419804464861581908681857, 0.29617908706340276564080698536, 1.45776270866997950108013675034, 3.42010647005122127951423262173, 4.43186894267887498358314053488, 5.01300215354758411114108766862, 6.04149279134545352737000582685, 6.62952838870393028636565816863, 7.61496414265307545640756963918, 7.86614763824635975744495654515, 8.573274284387035547990493789044

Graph of the ZZ-function along the critical line