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Properties

Label 2-5225-1.1-c1-0-76
Degree 22
Conductor 52255225
Sign 11
Analytic cond. 41.721841.7218
Root an. cond. 6.459246.45924
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  − 2.58·2-s − 1.69·3-s + 4.68·4-s + 4.38·6-s − 2.28·7-s − 6.95·8-s − 0.129·9-s + 11-s − 7.94·12-s + 1.76·13-s + 5.91·14-s + 8.60·16-s + 7.54·17-s + 0.333·18-s − 19-s + 3.87·21-s − 2.58·22-s + 6.89·23-s + 11.7·24-s − 4.56·26-s + 5.30·27-s − 10.7·28-s + 6.12·29-s + 8.39·31-s − 8.34·32-s − 1.69·33-s − 19.5·34-s + ⋯
L(s)  = 1  − 1.82·2-s − 0.978·3-s + 2.34·4-s + 1.78·6-s − 0.864·7-s − 2.45·8-s − 0.0430·9-s + 0.301·11-s − 2.29·12-s + 0.489·13-s + 1.58·14-s + 2.15·16-s + 1.83·17-s + 0.0787·18-s − 0.229·19-s + 0.846·21-s − 0.551·22-s + 1.43·23-s + 2.40·24-s − 0.894·26-s + 1.02·27-s − 2.02·28-s + 1.13·29-s + 1.50·31-s − 1.47·32-s − 0.294·33-s − 3.34·34-s + ⋯

Functional equation

Λ(s)=(5225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52255225    =    52⋅11⋅195^{2} \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 41.721841.7218
Root analytic conductor: 6.459246.45924
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5225, ( :1/2), 1)(2,\ 5225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.59305220330.5930522033
L(12)L(\frac12) ≈\approx 0.59305220330.5930522033
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−T 1 - T
19 1+T 1 + T
good2 1+2.58T+2T2 1 + 2.58T + 2T^{2}
3 1+1.69T+3T2 1 + 1.69T + 3T^{2}
7 1+2.28T+7T2 1 + 2.28T + 7T^{2}
13 1−1.76T+13T2 1 - 1.76T + 13T^{2}
17 1−7.54T+17T2 1 - 7.54T + 17T^{2}
23 1−6.89T+23T2 1 - 6.89T + 23T^{2}
29 1−6.12T+29T2 1 - 6.12T + 29T^{2}
31 1−8.39T+31T2 1 - 8.39T + 31T^{2}
37 1−10.1T+37T2 1 - 10.1T + 37T^{2}
41 1−4.07T+41T2 1 - 4.07T + 41T^{2}
43 1−5.51T+43T2 1 - 5.51T + 43T^{2}
47 1+11.6T+47T2 1 + 11.6T + 47T^{2}
53 1+12.7T+53T2 1 + 12.7T + 53T^{2}
59 1−13.5T+59T2 1 - 13.5T + 59T^{2}
61 1−5.89T+61T2 1 - 5.89T + 61T^{2}
67 1−8.79T+67T2 1 - 8.79T + 67T^{2}
71 1+9.61T+71T2 1 + 9.61T + 71T^{2}
73 1+8.70T+73T2 1 + 8.70T + 73T^{2}
79 1−4.40T+79T2 1 - 4.40T + 79T^{2}
83 1+0.146T+83T2 1 + 0.146T + 83T^{2}
89 1+1.54T+89T2 1 + 1.54T + 89T^{2}
97 1−7.32T+97T2 1 - 7.32T + 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

−8.254281641584486756833128853181, −7.66770571685439936105040872873, −6.69530401333914072316302565304, −6.37470430721832913284404163632, −5.74436056248877882741328615991, −4.71150173019947864796644406494, −3.25932753888294332064947660479, −2.70857259941319415596866916510, −1.15308681206483535667537183174, −0.71538213076460783716251036149, 0.71538213076460783716251036149, 1.15308681206483535667537183174, 2.70857259941319415596866916510, 3.25932753888294332064947660479, 4.71150173019947864796644406494, 5.74436056248877882741328615991, 6.37470430721832913284404163632, 6.69530401333914072316302565304, 7.66770571685439936105040872873, 8.254281641584486756833128853181

Graph of the ZZ-function along the critical line