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Properties

Label 2-4275-1.1-c1-0-108
Degree 22
Conductor 42754275
Sign −1-1
Analytic cond. 34.136034.1360
Root an. cond. 5.842605.84260
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.246·2-s − 1.93·4-s + 1.69·7-s − 0.972·8-s + 0.911·11-s + 1.55·13-s + 0.417·14-s + 3.63·16-s − 5.29·17-s − 19-s + 0.225·22-s − 4.24·23-s + 0.384·26-s − 3.28·28-s − 5.00·29-s + 1.82·31-s + 2.84·32-s − 1.30·34-s + 6.29·37-s − 0.246·38-s − 4.18·41-s + 7.31·43-s − 1.76·44-s − 1.04·46-s + 2.04·47-s − 4.13·49-s − 3.01·52-s + ⋯
L(s)  = 1  + 0.174·2-s − 0.969·4-s + 0.639·7-s − 0.343·8-s + 0.274·11-s + 0.431·13-s + 0.111·14-s + 0.909·16-s − 1.28·17-s − 0.229·19-s + 0.0480·22-s − 0.885·23-s + 0.0753·26-s − 0.620·28-s − 0.930·29-s + 0.328·31-s + 0.502·32-s − 0.224·34-s + 1.03·37-s − 0.0400·38-s − 0.652·41-s + 1.11·43-s − 0.266·44-s − 0.154·46-s + 0.298·47-s − 0.591·49-s − 0.418·52-s + ⋯

Functional equation

Λ(s)=(4275s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4275s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42754275    =    32⋅52⋅193^{2} \cdot 5^{2} \cdot 19
Sign: −1-1
Analytic conductor: 34.136034.1360
Root analytic conductor: 5.842605.84260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4275, ( :1/2), −1)(2,\ 4275,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
5 1 1
19 1+T 1 + T
good2 1−0.246T+2T2 1 - 0.246T + 2T^{2}
7 1−1.69T+7T2 1 - 1.69T + 7T^{2}
11 1−0.911T+11T2 1 - 0.911T + 11T^{2}
13 1−1.55T+13T2 1 - 1.55T + 13T^{2}
17 1+5.29T+17T2 1 + 5.29T + 17T^{2}
23 1+4.24T+23T2 1 + 4.24T + 23T^{2}
29 1+5.00T+29T2 1 + 5.00T + 29T^{2}
31 1−1.82T+31T2 1 - 1.82T + 31T^{2}
37 1−6.29T+37T2 1 - 6.29T + 37T^{2}
41 1+4.18T+41T2 1 + 4.18T + 41T^{2}
43 1−7.31T+43T2 1 - 7.31T + 43T^{2}
47 1−2.04T+47T2 1 - 2.04T + 47T^{2}
53 1−2.70T+53T2 1 - 2.70T + 53T^{2}
59 1+9.87T+59T2 1 + 9.87T + 59T^{2}
61 1−0.542T+61T2 1 - 0.542T + 61T^{2}
67 1+13.9T+67T2 1 + 13.9T + 67T^{2}
71 1−12.8T+71T2 1 - 12.8T + 71T^{2}
73 1+2.80T+73T2 1 + 2.80T + 73T^{2}
79 1−1.59T+79T2 1 - 1.59T + 79T^{2}
83 1+12.2T+83T2 1 + 12.2T + 83T^{2}
89 1+2.91T+89T2 1 + 2.91T + 89T^{2}
97 1−1.55T+97T2 1 - 1.55T + 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

−8.107194304989024981418223306551, −7.46082487381810938075252772941, −6.36085002128734901545080700788, −5.82015000606447604151477099607, −4.85016213006715481697956828238, −4.30285423089235161442316339831, −3.64630473049700540338608358491, −2.43657219240356756124696537364, −1.35707431756539852706439668830, 0, 1.35707431756539852706439668830, 2.43657219240356756124696537364, 3.64630473049700540338608358491, 4.30285423089235161442316339831, 4.85016213006715481697956828238, 5.82015000606447604151477099607, 6.36085002128734901545080700788, 7.46082487381810938075252772941, 8.107194304989024981418223306551

Graph of the ZZ-function along the critical line