Properties

Label 2-418-1.1-c1-0-3
Degree $2$
Conductor $418$
Sign $-1$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

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

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

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
11 \( 1 - T \)
19 \( 1 - T \)
good3 \( 1 + 3.30T + 3T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
7 \( 1 + 2.30T + 7T^{2} \)
13 \( 1 - 0.302T + 13T^{2} \)
17 \( 1 - 2.60T + 17T^{2} \)
23 \( 1 + 8.60T + 23T^{2} \)
29 \( 1 + 4.69T + 29T^{2} \)
31 \( 1 - 0.302T + 31T^{2} \)
37 \( 1 + 9.21T + 37T^{2} \)
41 \( 1 + 6.90T + 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 3.39T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 3.21T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 4.60T + 73T^{2} \)
79 \( 1 + 5.81T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + 2.60T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line