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Properties

Label 2-4205-1.1-c1-0-158
Degree 22
Conductor 42054205
Sign −1-1
Analytic cond. 33.577033.5770
Root an. cond. 5.794575.79457
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  + 0.517·2-s − 1.41·3-s − 1.73·4-s + 5-s − 0.732·6-s + 0.732·7-s − 1.93·8-s − 0.999·9-s + 0.517·10-s − 5.27·11-s + 2.44·12-s + 1.46·13-s + 0.378·14-s − 1.41·15-s + 2.46·16-s + 6.31·17-s − 0.517·18-s + 4.24·19-s − 1.73·20-s − 1.03·21-s − 2.73·22-s − 8.19·23-s + 2.73·24-s + 25-s + 0.757·26-s + 5.65·27-s − 1.26·28-s + ⋯
L(s)  = 1  + 0.366·2-s − 0.816·3-s − 0.866·4-s + 0.447·5-s − 0.298·6-s + 0.276·7-s − 0.683·8-s − 0.333·9-s + 0.163·10-s − 1.59·11-s + 0.707·12-s + 0.406·13-s + 0.101·14-s − 0.365·15-s + 0.616·16-s + 1.53·17-s − 0.122·18-s + 0.973·19-s − 0.387·20-s − 0.225·21-s − 0.582·22-s − 1.70·23-s + 0.557·24-s + 0.200·25-s + 0.148·26-s + 1.08·27-s − 0.239·28-s + ⋯

Functional equation

Λ(s)=(4205s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4205s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42054205    =    5⋅2925 \cdot 29^{2}
Sign: −1-1
Analytic conductor: 33.577033.5770
Root analytic conductor: 5.794575.79457
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4205, ( :1/2), −1)(2,\ 4205,\ (\ :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)
bad5 1−T 1 - T
29 1 1
good2 1−0.517T+2T2 1 - 0.517T + 2T^{2}
3 1+1.41T+3T2 1 + 1.41T + 3T^{2}
7 1−0.732T+7T2 1 - 0.732T + 7T^{2}
11 1+5.27T+11T2 1 + 5.27T + 11T^{2}
13 1−1.46T+13T2 1 - 1.46T + 13T^{2}
17 1−6.31T+17T2 1 - 6.31T + 17T^{2}
19 1−4.24T+19T2 1 - 4.24T + 19T^{2}
23 1+8.19T+23T2 1 + 8.19T + 23T^{2}
31 1−4.24T+31T2 1 - 4.24T + 31T^{2}
37 1−4.24T+37T2 1 - 4.24T + 37T^{2}
41 1+8.76T+41T2 1 + 8.76T + 41T^{2}
43 1−4.24T+43T2 1 - 4.24T + 43T^{2}
47 1−8.38T+47T2 1 - 8.38T + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1+3.10T+61T2 1 + 3.10T + 61T^{2}
67 1+11.1T+67T2 1 + 11.1T + 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+1.13T+73T2 1 + 1.13T + 73T^{2}
79 1+15.8T+79T2 1 + 15.8T + 79T^{2}
83 1−2.19T+83T2 1 - 2.19T + 83T^{2}
89 1−2.07T+89T2 1 - 2.07T + 89T^{2}
97 1+7.34T+97T2 1 + 7.34T + 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

−8.079138062372543684168792319056, −7.42710605202480751322336586539, −6.03437566868495528297008573110, −5.75943892175074466638455777451, −5.20250379698199247263715194308, −4.51308711054977261724012577154, −3.40661485173807791591545838114, −2.65548337108864378230450963148, −1.15983514543131789642637339590, 0, 1.15983514543131789642637339590, 2.65548337108864378230450963148, 3.40661485173807791591545838114, 4.51308711054977261724012577154, 5.20250379698199247263715194308, 5.75943892175074466638455777451, 6.03437566868495528297008573110, 7.42710605202480751322336586539, 8.079138062372543684168792319056

Graph of the ZZ-function along the critical line