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Properties

Label 2-2695-2695.879-c0-0-0
Degree 22
Conductor 26952695
Sign −0.718−0.695i-0.718 - 0.695i
Analytic cond. 1.344981.34498
Root an. cond. 1.159731.15973
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  + (−0.496 + 1.26i)2-s + (−0.623 − 0.578i)4-s + (−0.826 − 0.563i)5-s + (0.781 + 0.623i)7-s + (−0.183 + 0.0882i)8-s + (−0.988 − 0.149i)9-s + (1.12 − 0.766i)10-s + (0.988 − 0.149i)11-s + (0.185 + 0.233i)13-s + (−1.17 + 0.680i)14-s + (−0.0842 − 1.12i)16-s + (0.829 + 0.255i)17-s + (0.680 − 1.17i)18-s + (0.189 + 0.829i)20-s + (−0.302 + 1.32i)22-s + ⋯
L(s)  = 1  + (−0.496 + 1.26i)2-s + (−0.623 − 0.578i)4-s + (−0.826 − 0.563i)5-s + (0.781 + 0.623i)7-s + (−0.183 + 0.0882i)8-s + (−0.988 − 0.149i)9-s + (1.12 − 0.766i)10-s + (0.988 − 0.149i)11-s + (0.185 + 0.233i)13-s + (−1.17 + 0.680i)14-s + (−0.0842 − 1.12i)16-s + (0.829 + 0.255i)17-s + (0.680 − 1.17i)18-s + (0.189 + 0.829i)20-s + (−0.302 + 1.32i)22-s + ⋯

Functional equation

Λ(s)=(2695s/2 ΓC(s) L(s)=((−0.718−0.695i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2695s/2 ΓC(s) L(s)=((−0.718−0.695i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26952695    =    5⋅72⋅115 \cdot 7^{2} \cdot 11
Sign: −0.718−0.695i-0.718 - 0.695i
Analytic conductor: 1.344981.34498
Root analytic conductor: 1.159731.15973
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2695(879,⋅)\chi_{2695} (879, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2695, ( :0), −0.718−0.695i)(2,\ 2695,\ (\ :0),\ -0.718 - 0.695i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.78413365470.7841336547
L(12)L(\frac12) ≈\approx 0.78413365470.7841336547
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)
bad5 1+(0.826+0.563i)T 1 + (0.826 + 0.563i)T
7 1+(−0.781−0.623i)T 1 + (-0.781 - 0.623i)T
11 1+(−0.988+0.149i)T 1 + (-0.988 + 0.149i)T
good2 1+(0.496−1.26i)T+(−0.733−0.680i)T2 1 + (0.496 - 1.26i)T + (-0.733 - 0.680i)T^{2}
3 1+(0.988+0.149i)T2 1 + (0.988 + 0.149i)T^{2}
13 1+(−0.185−0.233i)T+(−0.222+0.974i)T2 1 + (-0.185 - 0.233i)T + (-0.222 + 0.974i)T^{2}
17 1+(−0.829−0.255i)T+(0.826+0.563i)T2 1 + (-0.829 - 0.255i)T + (0.826 + 0.563i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
29 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
31 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
41 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
43 1+(−1.67−0.807i)T+(0.623+0.781i)T2 1 + (-1.67 - 0.807i)T + (0.623 + 0.781i)T^{2}
47 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
53 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
59 1+(1.63−1.11i)T+(0.365−0.930i)T2 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2}
61 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(−0.326+1.42i)T+(−0.900−0.433i)T2 1 + (-0.326 + 1.42i)T + (-0.900 - 0.433i)T^{2}
73 1+(−0.728−1.85i)T+(−0.733+0.680i)T2 1 + (-0.728 - 1.85i)T + (-0.733 + 0.680i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(1.07−1.35i)T+(−0.222−0.974i)T2 1 + (1.07 - 1.35i)T + (-0.222 - 0.974i)T^{2}
89 1+(−0.722−0.108i)T+(0.955+0.294i)T2 1 + (-0.722 - 0.108i)T + (0.955 + 0.294i)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

−9.015876718739275069251284408156, −8.420169480782557111754960258682, −7.923191160619563044525493804799, −7.20826901231741039267182770211, −6.23754318165434744791557249853, −5.64890072029216468835447608492, −4.92376732702791931622739943492, −3.89060894563345141035451981089, −2.83925595087060949420068231606, −1.22416857843272225039711025874, 0.69857111474213093599801790275, 1.90597792739310737185477140372, 2.96763787266837347097950731164, 3.67632825674986544835017254663, 4.37780987389798907681202849478, 5.62203148522713636132939400721, 6.52920568139613575052875683193, 7.52039538294681336767129692881, 8.071765596379128686187035036100, 8.895108949436773114714943859409

Graph of the ZZ-function along the critical line