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Properties

Label 2-4675-1.1-c1-0-127
Degree 22
Conductor 46754675
Sign 11
Analytic cond. 37.330037.3300
Root an. cond. 6.109836.10983
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  + 1.21·2-s + 2.36·3-s − 0.515·4-s + 2.87·6-s + 0.846·7-s − 3.06·8-s + 2.58·9-s − 11-s − 1.21·12-s + 0.935·13-s + 1.03·14-s − 2.70·16-s + 17-s + 3.14·18-s + 5.75·19-s + 2·21-s − 1.21·22-s − 2.54·23-s − 7.24·24-s + 1.13·26-s − 0.990·27-s − 0.436·28-s + 9.45·29-s + 4.12·31-s + 2.83·32-s − 2.36·33-s + 1.21·34-s + ⋯
L(s)  = 1  + 0.861·2-s + 1.36·3-s − 0.257·4-s + 1.17·6-s + 0.319·7-s − 1.08·8-s + 0.860·9-s − 0.301·11-s − 0.351·12-s + 0.259·13-s + 0.275·14-s − 0.675·16-s + 0.242·17-s + 0.741·18-s + 1.32·19-s + 0.436·21-s − 0.259·22-s − 0.531·23-s − 1.47·24-s + 0.223·26-s − 0.190·27-s − 0.0825·28-s + 1.75·29-s + 0.741·31-s + 0.501·32-s − 0.411·33-s + 0.208·34-s + ⋯

Functional equation

Λ(s)=(4675s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4675s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46754675    =    52⋅11⋅175^{2} \cdot 11 \cdot 17
Sign: 11
Analytic conductor: 37.330037.3300
Root analytic conductor: 6.109836.10983
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4675, ( :1/2), 1)(2,\ 4675,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.5010013454.501001345
L(12)L(\frac12) ≈\approx 4.5010013454.501001345
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
17 1−T 1 - T
good2 1−1.21T+2T2 1 - 1.21T + 2T^{2}
3 1−2.36T+3T2 1 - 2.36T + 3T^{2}
7 1−0.846T+7T2 1 - 0.846T + 7T^{2}
13 1−0.935T+13T2 1 - 0.935T + 13T^{2}
19 1−5.75T+19T2 1 - 5.75T + 19T^{2}
23 1+2.54T+23T2 1 + 2.54T + 23T^{2}
29 1−9.45T+29T2 1 - 9.45T + 29T^{2}
31 1−4.12T+31T2 1 - 4.12T + 31T^{2}
37 1−0.776T+37T2 1 - 0.776T + 37T^{2}
41 1+2.54T+41T2 1 + 2.54T + 41T^{2}
43 1−12.5T+43T2 1 - 12.5T + 43T^{2}
47 1−3.81T+47T2 1 - 3.81T + 47T^{2}
53 1−7.94T+53T2 1 - 7.94T + 53T^{2}
59 1−11.8T+59T2 1 - 11.8T + 59T^{2}
61 1+9.72T+61T2 1 + 9.72T + 61T^{2}
67 1+1.13T+67T2 1 + 1.13T + 67T^{2}
71 1+6.87T+71T2 1 + 6.87T + 71T^{2}
73 1−9.13T+73T2 1 - 9.13T + 73T^{2}
79 1−10.8T+79T2 1 - 10.8T + 79T^{2}
83 1−9.10T+83T2 1 - 9.10T + 83T^{2}
89 1+0.835T+89T2 1 + 0.835T + 89T^{2}
97 1+8.21T+97T2 1 + 8.21T + 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

−8.227305018659894591812886978055, −7.84534792311009503447869099393, −6.87996653860080187659039300898, −5.93613894518723919906825109512, −5.22683762668709111034486538608, −4.41824722826831634403782856247, −3.73856900327122958065009695992, −2.97534975684993375382473718888, −2.42345514140626795396446306676, −1.00834730735439999590031028305, 1.00834730735439999590031028305, 2.42345514140626795396446306676, 2.97534975684993375382473718888, 3.73856900327122958065009695992, 4.41824722826831634403782856247, 5.22683762668709111034486538608, 5.93613894518723919906825109512, 6.87996653860080187659039300898, 7.84534792311009503447869099393, 8.227305018659894591812886978055

Graph of the ZZ-function along the critical line