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Properties

Label 2-2695-2695.2584-c0-0-3
Degree 22
Conductor 26952695
Sign 0.462−0.886i0.462 - 0.886i
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  + (0.193 + 0.846i)2-s + (0.222 − 0.107i)4-s + (−0.623 − 0.781i)5-s + (0.974 + 0.222i)7-s + (0.674 + 0.846i)8-s + (−0.222 + 0.974i)9-s + (0.541 − 0.678i)10-s + (0.222 + 0.974i)11-s + (−0.433 − 1.90i)13-s + 0.867i·14-s + (−0.431 + 0.541i)16-s + (1.40 + 0.678i)17-s − 0.867·18-s + (−0.222 − 0.107i)20-s + (−0.781 + 0.376i)22-s + ⋯
L(s)  = 1  + (0.193 + 0.846i)2-s + (0.222 − 0.107i)4-s + (−0.623 − 0.781i)5-s + (0.974 + 0.222i)7-s + (0.674 + 0.846i)8-s + (−0.222 + 0.974i)9-s + (0.541 − 0.678i)10-s + (0.222 + 0.974i)11-s + (−0.433 − 1.90i)13-s + 0.867i·14-s + (−0.431 + 0.541i)16-s + (1.40 + 0.678i)17-s − 0.867·18-s + (−0.222 − 0.107i)20-s + (−0.781 + 0.376i)22-s + ⋯

Functional equation

Λ(s)=(2695s/2 ΓC(s) L(s)=((0.462−0.886i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2695s/2 ΓC(s) L(s)=((0.462−0.886i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26952695    =    5⋅72⋅115 \cdot 7^{2} \cdot 11
Sign: 0.462−0.886i0.462 - 0.886i
Analytic conductor: 1.344981.34498
Root analytic conductor: 1.159731.15973
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2695(2584,⋅)\chi_{2695} (2584, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2695, ( :0), 0.462−0.886i)(2,\ 2695,\ (\ :0),\ 0.462 - 0.886i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6058050981.605805098
L(12)L(\frac12) ≈\approx 1.6058050981.605805098
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+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
7 1+(−0.974−0.222i)T 1 + (-0.974 - 0.222i)T
11 1+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
good2 1+(−0.193−0.846i)T+(−0.900+0.433i)T2 1 + (-0.193 - 0.846i)T + (-0.900 + 0.433i)T^{2}
3 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
13 1+(0.433+1.90i)T+(−0.900+0.433i)T2 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2}
17 1+(−1.40−0.678i)T+(0.623+0.781i)T2 1 + (-1.40 - 0.678i)T + (0.623 + 0.781i)T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
29 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
31 1+2T+T2 1 + 2T + T^{2}
37 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
41 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
43 1+(−1.21+1.52i)T+(−0.222−0.974i)T2 1 + (-1.21 + 1.52i)T + (-0.222 - 0.974i)T^{2}
47 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
53 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
59 1+(0.277−0.347i)T+(−0.222−0.974i)T2 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2}
61 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−1.62+0.781i)T+(0.623−0.781i)T2 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2}
73 1+(0.347−1.52i)T+(−0.900−0.433i)T2 1 + (0.347 - 1.52i)T + (-0.900 - 0.433i)T^{2}
79 1−T2 1 - T^{2}
83 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
89 1+(0.0990−0.433i)T+(−0.900−0.433i)T2 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)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

−8.763545937757817608504459275832, −8.063460461137399024774372235727, −7.62897326907777891151978434935, −7.28165533929102627828688190310, −5.62910206702146335522453625094, −5.45642088636639488650001204696, −4.83326143791814100884173319461, −3.78699706385628723020671909674, −2.38180954949052421707082172288, −1.42272115654141410226558327340, 1.15297601978162516498118864847, 2.26872439675699122113991690909, 3.31455803751174280386847312759, 3.80533413011371039449391189102, 4.62472237486082719501214007399, 5.88269617058099557546527992106, 6.77332971620844935882865908826, 7.34079131916835929851508694474, 7.994011890078519224331911732448, 9.112109645667362200632082467548

Graph of the ZZ-function along the critical line