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Properties

Label 2-418-1.1-c1-0-3
Degree 22
Conductor 418418
Sign −1-1
Analytic cond. 3.337743.33774
Root an. cond. 1.826951.82695
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  − 2-s − 3.30·3-s + 4-s + 1.30·5-s + 3.30·6-s − 2.30·7-s − 8-s + 7.90·9-s − 1.30·10-s + 11-s − 3.30·12-s + 0.302·13-s + 2.30·14-s − 4.30·15-s + 16-s + 2.60·17-s − 7.90·18-s + 19-s + 1.30·20-s + 7.60·21-s − 22-s − 8.60·23-s + 3.30·24-s − 3.30·25-s − 0.302·26-s − 16.2·27-s − 2.30·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.90·3-s + 0.5·4-s + 0.582·5-s + 1.34·6-s − 0.870·7-s − 0.353·8-s + 2.63·9-s − 0.411·10-s + 0.301·11-s − 0.953·12-s + 0.0839·13-s + 0.615·14-s − 1.11·15-s + 0.250·16-s + 0.631·17-s − 1.86·18-s + 0.229·19-s + 0.291·20-s + 1.65·21-s − 0.213·22-s − 1.79·23-s + 0.674·24-s − 0.660·25-s − 0.0593·26-s − 3.11·27-s − 0.435·28-s + ⋯

Functional equation

Λ(s)=(418s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(418s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 418418    =    2⋅11⋅192 \cdot 11 \cdot 19
Sign: −1-1
Analytic conductor: 3.337743.33774
Root analytic conductor: 1.826951.82695
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 418, ( :1/2), −1)(2,\ 418,\ (\ :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)
bad2 1+T 1 + T
11 1−T 1 - T
19 1−T 1 - T
good3 1+3.30T+3T2 1 + 3.30T + 3T^{2}
5 1−1.30T+5T2 1 - 1.30T + 5T^{2}
7 1+2.30T+7T2 1 + 2.30T + 7T^{2}
13 1−0.302T+13T2 1 - 0.302T + 13T^{2}
17 1−2.60T+17T2 1 - 2.60T + 17T^{2}
23 1+8.60T+23T2 1 + 8.60T + 23T^{2}
29 1+4.69T+29T2 1 + 4.69T + 29T^{2}
31 1−0.302T+31T2 1 - 0.302T + 31T^{2}
37 1+9.21T+37T2 1 + 9.21T + 37T^{2}
41 1+6.90T+41T2 1 + 6.90T + 41T^{2}
43 1−11.9T+43T2 1 - 11.9T + 43T^{2}
47 1+6T+47T2 1 + 6T + 47T^{2}
53 1+3.39T+53T2 1 + 3.39T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+3.21T+61T2 1 + 3.21T + 61T^{2}
67 1−11.5T+67T2 1 - 11.5T + 67T^{2}
71 1+13.3T+71T2 1 + 13.3T + 71T^{2}
73 1−4.60T+73T2 1 - 4.60T + 73T^{2}
79 1+5.81T+79T2 1 + 5.81T + 79T^{2}
83 1−10.6T+83T2 1 - 10.6T + 83T^{2}
89 1+2.60T+89T2 1 + 2.60T + 89T^{2}
97 1−8T+97T2 1 - 8T + 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

−10.60287725259275182621730642620, −9.986012943794927514747100801308, −9.407947924227090552040224529809, −7.76961005975310623714196819576, −6.72763010216460240577597558624, −6.06357461214933637977396780043, −5.37952484435793036848315259052, −3.83530467209852321084430827197, −1.65980106514964114310882374667, 0, 1.65980106514964114310882374667, 3.83530467209852321084430827197, 5.37952484435793036848315259052, 6.06357461214933637977396780043, 6.72763010216460240577597558624, 7.76961005975310623714196819576, 9.407947924227090552040224529809, 9.986012943794927514747100801308, 10.60287725259275182621730642620

Graph of the ZZ-function along the critical line