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Properties

Label 2-845-1.1-c1-0-37
Degree 22
Conductor 845845
Sign 11
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s + 3-s + 3.30·4-s + 5-s + 2.30·6-s + 7-s + 3.00·8-s − 2·9-s + 2.30·10-s − 1.60·11-s + 3.30·12-s + 2.30·14-s + 15-s + 0.302·16-s + 7.60·17-s − 4.60·18-s + 5.60·19-s + 3.30·20-s + 21-s − 3.69·22-s − 3·23-s + 3.00·24-s + 25-s − 5·27-s + 3.30·28-s − 6.21·29-s + 2.30·30-s + ⋯
L(s)  = 1  + 1.62·2-s + 0.577·3-s + 1.65·4-s + 0.447·5-s + 0.940·6-s + 0.377·7-s + 1.06·8-s − 0.666·9-s + 0.728·10-s − 0.484·11-s + 0.953·12-s + 0.615·14-s + 0.258·15-s + 0.0756·16-s + 1.84·17-s − 1.08·18-s + 1.28·19-s + 0.738·20-s + 0.218·21-s − 0.788·22-s − 0.625·23-s + 0.612·24-s + 0.200·25-s − 0.962·27-s + 0.624·28-s − 1.15·29-s + 0.420·30-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 11
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), 1)(2,\ 845,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.7412364324.741236432
L(12)L(\frac12) ≈\approx 4.7412364324.741236432
L(32)L(\frac{3}{2}) 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−T 1 - T
13 1 1
good2 1−2.30T+2T2 1 - 2.30T + 2T^{2}
3 1−T+3T2 1 - T + 3T^{2}
7 1−T+7T2 1 - T + 7T^{2}
11 1+1.60T+11T2 1 + 1.60T + 11T^{2}
17 1−7.60T+17T2 1 - 7.60T + 17T^{2}
19 1−5.60T+19T2 1 - 5.60T + 19T^{2}
23 1+3T+23T2 1 + 3T + 23T^{2}
29 1+6.21T+29T2 1 + 6.21T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+3.60T+37T2 1 + 3.60T + 37T^{2}
41 1+3T+41T2 1 + 3T + 41T^{2}
43 1+10.2T+43T2 1 + 10.2T + 43T^{2}
47 1+9.21T+47T2 1 + 9.21T + 47T^{2}
53 1+3.21T+53T2 1 + 3.21T + 53T^{2}
59 1−10.8T+59T2 1 - 10.8T + 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1−7T+67T2 1 - 7T + 67T^{2}
71 1+4.81T+71T2 1 + 4.81T + 71T^{2}
73 1−0.788T+73T2 1 - 0.788T + 73T^{2}
79 1−5.21T+79T2 1 - 5.21T + 79T^{2}
83 1−9.21T+83T2 1 - 9.21T + 83T^{2}
89 1−6.21T+89T2 1 - 6.21T + 89T^{2}
97 1−8.39T+97T2 1 - 8.39T + 97T^{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

−10.23948127815180831698568867260, −9.491597028274481016595490014246, −8.249182346494674831196633259830, −7.58170703178011570551686526041, −6.39699950830088178096986400096, −5.41123508753474257536257669286, −5.11624718349095016440146693414, −3.56770180873369175370136759490, −3.08475022846135840260877917843, −1.86263334625763701777694191280, 1.86263334625763701777694191280, 3.08475022846135840260877917843, 3.56770180873369175370136759490, 5.11624718349095016440146693414, 5.41123508753474257536257669286, 6.39699950830088178096986400096, 7.58170703178011570551686526041, 8.249182346494674831196633259830, 9.491597028274481016595490014246, 10.23948127815180831698568867260

Graph of the ZZ-function along the critical line