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Properties

Label 2-2695-2695.1044-c0-0-3
Degree 22
Conductor 26952695
Sign −0.462−0.886i-0.462 - 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  + (0.400 − 1.75i)2-s + (−2.02 − 0.974i)4-s + (0.623 − 0.781i)5-s + (−0.222 − 0.974i)7-s + (−1.40 + 1.75i)8-s + (−0.222 − 0.974i)9-s + (−1.12 − 1.40i)10-s + (−0.222 + 0.974i)11-s + (0.0990 − 0.433i)13-s − 1.80·14-s + (1.12 + 1.40i)16-s + (−1.12 + 0.541i)17-s − 1.80·18-s + (−2.02 + 0.974i)20-s + (1.62 + 0.781i)22-s + ⋯
L(s)  = 1  + (0.400 − 1.75i)2-s + (−2.02 − 0.974i)4-s + (0.623 − 0.781i)5-s + (−0.222 − 0.974i)7-s + (−1.40 + 1.75i)8-s + (−0.222 − 0.974i)9-s + (−1.12 − 1.40i)10-s + (−0.222 + 0.974i)11-s + (0.0990 − 0.433i)13-s − 1.80·14-s + (1.12 + 1.40i)16-s + (−1.12 + 0.541i)17-s − 1.80·18-s + (−2.02 + 0.974i)20-s + (1.62 + 0.781i)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: −0.462−0.886i-0.462 - 0.886i
Analytic conductor: 1.344981.34498
Root analytic conductor: 1.159731.15973
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2695(1044,⋅)\chi_{2695} (1044, \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 1.1752768331.175276833
L(12)L(\frac12) ≈\approx 1.1752768331.175276833
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+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
11 1+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
good2 1+(−0.400+1.75i)T+(−0.900−0.433i)T2 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2}
3 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
13 1+(−0.0990+0.433i)T+(−0.900−0.433i)T2 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2}
17 1+(1.12−0.541i)T+(0.623−0.781i)T2 1 + (1.12 - 0.541i)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+(0.277+0.347i)T+(−0.222+0.974i)T2 1 + (0.277 + 0.347i)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+(0.277+1.21i)T+(−0.900+0.433i)T2 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2}
79 1−T2 1 - T^{2}
83 1+(0.445+1.94i)T+(−0.900+0.433i)T2 1 + (0.445 + 1.94i)T + (-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

−8.907813485557227206725484982560, −8.105446911156888602132612842448, −6.79490691067171823247071494875, −6.02739009899954196266417031280, −4.88111949697985815194767182531, −4.41453483732496577930761410258, −3.63213090052810381425084531968, −2.63030301991671303054168064457, −1.66666898889664240597976311163, −0.66420208121727545364211250261, 2.32093845688993261484988632149, 3.11736329510512747983408362564, 4.42740974241054726342716054016, 5.20637601679095507984891318945, 5.83305309517963121809262169677, 6.45554149769377218489736544849, 6.97150096717538988770257263991, 8.020793456000352868939736123277, 8.482436954345827448653835620565, 9.190505982028224704526152455336

Graph of the ZZ-function along the critical line