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Properties

Label 2-55e2-1.1-c1-0-19
Degree 22
Conductor 30253025
Sign 11
Analytic cond. 24.154724.1547
Root an. cond. 4.914744.91474
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  + 1.65·2-s − 1.97·3-s + 0.737·4-s − 3.26·6-s − 2.24·7-s − 2.08·8-s + 0.899·9-s − 1.45·12-s − 3.69·13-s − 3.71·14-s − 4.93·16-s − 2.22·17-s + 1.48·18-s + 5.28·19-s + 4.42·21-s − 3.85·23-s + 4.12·24-s − 6.12·26-s + 4.14·27-s − 1.65·28-s + 0.188·29-s + 0.686·31-s − 3.98·32-s − 3.68·34-s + 0.663·36-s + 2.59·37-s + 8.74·38-s + ⋯
L(s)  = 1  + 1.16·2-s − 1.14·3-s + 0.368·4-s − 1.33·6-s − 0.847·7-s − 0.738·8-s + 0.299·9-s − 0.420·12-s − 1.02·13-s − 0.991·14-s − 1.23·16-s − 0.539·17-s + 0.350·18-s + 1.21·19-s + 0.966·21-s − 0.803·23-s + 0.841·24-s − 1.20·26-s + 0.798·27-s − 0.312·28-s + 0.0349·29-s + 0.123·31-s − 0.703·32-s − 0.631·34-s + 0.110·36-s + 0.426·37-s + 1.41·38-s + ⋯

Functional equation

Λ(s)=(3025s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3025s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30253025    =    52⋅1125^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 24.154724.1547
Root analytic conductor: 4.914744.91474
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3025, ( :1/2), 1)(2,\ 3025,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1093682321.109368232
L(12)L(\frac12) ≈\approx 1.1093682321.109368232
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 1
11 1 1
good2 1−1.65T+2T2 1 - 1.65T + 2T^{2}
3 1+1.97T+3T2 1 + 1.97T + 3T^{2}
7 1+2.24T+7T2 1 + 2.24T + 7T^{2}
13 1+3.69T+13T2 1 + 3.69T + 13T^{2}
17 1+2.22T+17T2 1 + 2.22T + 17T^{2}
19 1−5.28T+19T2 1 - 5.28T + 19T^{2}
23 1+3.85T+23T2 1 + 3.85T + 23T^{2}
29 1−0.188T+29T2 1 - 0.188T + 29T^{2}
31 1−0.686T+31T2 1 - 0.686T + 31T^{2}
37 1−2.59T+37T2 1 - 2.59T + 37T^{2}
41 1−7.91T+41T2 1 - 7.91T + 41T^{2}
43 1−8.41T+43T2 1 - 8.41T + 43T^{2}
47 1+12.0T+47T2 1 + 12.0T + 47T^{2}
53 1−12.6T+53T2 1 - 12.6T + 53T^{2}
59 1+0.343T+59T2 1 + 0.343T + 59T^{2}
61 1−1.73T+61T2 1 - 1.73T + 61T^{2}
67 1−0.650T+67T2 1 - 0.650T + 67T^{2}
71 1−4.64T+71T2 1 - 4.64T + 71T^{2}
73 1+8.85T+73T2 1 + 8.85T + 73T^{2}
79 1+7.23T+79T2 1 + 7.23T + 79T^{2}
83 1−3.18T+83T2 1 - 3.18T + 83T^{2}
89 1−9.92T+89T2 1 - 9.92T + 89T^{2}
97 1−2.26T+97T2 1 - 2.26T + 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

−8.870146286628711338609912110373, −7.66811660495990753831464835826, −6.84360758364492796409815979189, −6.15397960854221345059547004373, −5.63821408717182862189975139252, −4.92227875479514850636791271142, −4.24511460275724949214529443714, −3.25415728288556989093670153317, −2.43566478594833706154381027098, −0.53999591252574595235015351304, 0.53999591252574595235015351304, 2.43566478594833706154381027098, 3.25415728288556989093670153317, 4.24511460275724949214529443714, 4.92227875479514850636791271142, 5.63821408717182862189975139252, 6.15397960854221345059547004373, 6.84360758364492796409815979189, 7.66811660495990753831464835826, 8.870146286628711338609912110373

Graph of the ZZ-function along the critical line