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Properties

Label 2-2695-2695.1264-c0-0-2
Degree 22
Conductor 26952695
Sign 0.718−0.695i0.718 - 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  + (1.88 + 0.284i)2-s + (2.53 + 0.781i)4-s + (0.0747 + 0.997i)5-s + (−0.623 − 0.781i)7-s + (2.84 + 1.37i)8-s + (0.365 − 0.930i)9-s + (−0.142 + 1.90i)10-s + (0.365 + 0.930i)11-s + (−0.455 + 0.571i)13-s + (−0.955 − 1.65i)14-s + (2.79 + 1.90i)16-s + (−1.32 − 1.22i)17-s + (0.955 − 1.65i)18-s + (−0.590 + 2.58i)20-s + (0.425 + 1.86i)22-s + ⋯
L(s)  = 1  + (1.88 + 0.284i)2-s + (2.53 + 0.781i)4-s + (0.0747 + 0.997i)5-s + (−0.623 − 0.781i)7-s + (2.84 + 1.37i)8-s + (0.365 − 0.930i)9-s + (−0.142 + 1.90i)10-s + (0.365 + 0.930i)11-s + (−0.455 + 0.571i)13-s + (−0.955 − 1.65i)14-s + (2.79 + 1.90i)16-s + (−1.32 − 1.22i)17-s + (0.955 − 1.65i)18-s + (−0.590 + 2.58i)20-s + (0.425 + 1.86i)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: 0.718−0.695i0.718 - 0.695i
Analytic conductor: 1.344981.34498
Root analytic conductor: 1.159731.15973
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2695(1264,⋅)\chi_{2695} (1264, \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 3.7177195033.717719503
L(12)L(\frac12) ≈\approx 3.7177195033.717719503
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+(−0.0747−0.997i)T 1 + (-0.0747 - 0.997i)T
7 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
11 1+(−0.365−0.930i)T 1 + (-0.365 - 0.930i)T
good2 1+(−1.88−0.284i)T+(0.955+0.294i)T2 1 + (-1.88 - 0.284i)T + (0.955 + 0.294i)T^{2}
3 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
13 1+(0.455−0.571i)T+(−0.222−0.974i)T2 1 + (0.455 - 0.571i)T + (-0.222 - 0.974i)T^{2}
17 1+(1.32+1.22i)T+(0.0747+0.997i)T2 1 + (1.32 + 1.22i)T + (0.0747 + 0.997i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)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+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
41 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
43 1+(1.78−0.858i)T+(0.623−0.781i)T2 1 + (1.78 - 0.858i)T + (0.623 - 0.781i)T^{2}
47 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
53 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
59 1+(−0.0546+0.728i)T+(−0.988−0.149i)T2 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2}
61 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(0.425+1.86i)T+(−0.900+0.433i)T2 1 + (0.425 + 1.86i)T + (-0.900 + 0.433i)T^{2}
73 1+(−1.63+0.246i)T+(0.955−0.294i)T2 1 + (-1.63 + 0.246i)T + (0.955 - 0.294i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(−0.623−0.781i)T+(−0.222+0.974i)T2 1 + (-0.623 - 0.781i)T + (-0.222 + 0.974i)T^{2}
89 1+(0.722−1.84i)T+(−0.733−0.680i)T2 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)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

−9.424032637345551973859517560885, −7.74757345638506957335557577767, −7.08333171457563283089778138660, −6.52522053441582027905092448905, −6.44520549275085876620014555915, −4.98839888120436212650780441956, −4.32736100716790466135109822015, −3.68952737293232065767468327150, −2.87021352539814799554379538969, −1.99561542600636553158483207716, 1.63126418458918919616004466791, 2.45171403427100903590569369929, 3.43203423268548854165984110148, 4.26690701340183357443775992708, 5.02529617027298167675602694040, 5.59221013494507281117986730606, 6.27483089073410796940817337262, 7.00153926577918254218642432809, 8.208946612744423351956174304086, 8.759190690588576113204841214304

Graph of the ZZ-function along the critical line