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Properties

Label 2-2695-2695.2199-c0-0-1
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  + (0.277 − 0.347i)2-s + (0.178 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (0.722 + 0.347i)8-s + (0.623 + 0.781i)9-s + (−0.400 + 0.193i)10-s + (0.623 − 0.781i)11-s + (−0.777 + 0.974i)13-s − 0.445·14-s + (−0.400 + 0.193i)16-s + (−0.400 + 1.75i)17-s + 0.445·18-s + (0.178 − 0.781i)20-s + (−0.0990 − 0.433i)22-s + ⋯
L(s)  = 1  + (0.277 − 0.347i)2-s + (0.178 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (0.722 + 0.347i)8-s + (0.623 + 0.781i)9-s + (−0.400 + 0.193i)10-s + (0.623 − 0.781i)11-s + (−0.777 + 0.974i)13-s − 0.445·14-s + (−0.400 + 0.193i)16-s + (−0.400 + 1.75i)17-s + 0.445·18-s + (0.178 − 0.781i)20-s + (−0.0990 − 0.433i)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(2199,⋅)\chi_{2695} (2199, \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 1.1758690601.175869060
L(12)L(\frac12) ≈\approx 1.1758690601.175869060
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+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
7 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
11 1+(−0.623+0.781i)T 1 + (-0.623 + 0.781i)T
good2 1+(−0.277+0.347i)T+(−0.222−0.974i)T2 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2}
3 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
13 1+(0.777−0.974i)T+(−0.222−0.974i)T2 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2}
17 1+(0.400−1.75i)T+(−0.900−0.433i)T2 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2}
19 1−T2 1 - T^{2}
23 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
29 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
31 1−2T+T2 1 - 2T + T^{2}
37 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
41 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
43 1+(−1.12+0.541i)T+(0.623−0.781i)T2 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2}
47 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
53 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
59 1+(1.12−0.541i)T+(0.623−0.781i)T2 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2}
61 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−0.0990−0.433i)T+(−0.900+0.433i)T2 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2}
73 1+(−1.12−1.40i)T+(−0.222+0.974i)T2 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2}
79 1−T2 1 - T^{2}
83 1+(1.24+1.56i)T+(−0.222+0.974i)T2 1 + (1.24 + 1.56i)T + (-0.222 + 0.974i)T^{2}
89 1+(−0.777−0.974i)T+(−0.222+0.974i)T2 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)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

−8.927793766886312778189398033560, −8.285603827729393490434546711148, −7.63244985937952854486061467215, −6.96225857761534994001739993773, −6.24797865305195989263023827064, −4.76028654655301021609149678954, −4.15054059571389306922587230181, −3.77767644430200124063903982883, −2.63470544106479380627898913012, −1.40198919508518554901468866461, 0.75108509298949270153101790115, 2.41650450500442904162714123508, 3.21064316013800316981377161928, 4.43964978767609485227714749085, 4.89345220088770286613774120412, 6.02837591343826006609039134046, 6.70737277894111431551110092448, 7.16067552873967790007950905887, 7.931372513926666884041723509223, 9.184120293091039907927906607076

Graph of the ZZ-function along the critical line