Properties

Label 2-418-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.16·3-s + 4-s + 4.16·5-s − 2.16·6-s − 0.683·7-s + 8-s + 1.68·9-s + 4.16·10-s − 11-s − 2.16·12-s + 2.68·13-s − 0.683·14-s − 9.01·15-s + 16-s + 3.36·17-s + 1.68·18-s − 19-s + 4.16·20-s + 1.48·21-s − 22-s − 1.36·23-s − 2.16·24-s + 12.3·25-s + 2.68·26-s + 2.84·27-s − 0.683·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.24·3-s + 0.5·4-s + 1.86·5-s − 0.883·6-s − 0.258·7-s + 0.353·8-s + 0.561·9-s + 1.31·10-s − 0.301·11-s − 0.624·12-s + 0.744·13-s − 0.182·14-s − 2.32·15-s + 0.250·16-s + 0.816·17-s + 0.396·18-s − 0.229·19-s + 0.931·20-s + 0.323·21-s − 0.213·22-s − 0.285·23-s − 0.441·24-s + 2.46·25-s + 0.526·26-s + 0.548·27-s − 0.129·28-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: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.872909514\)
\(L(\frac12)\) \(\approx\) \(1.872909514\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.16T + 3T^{2} \)
5 \( 1 - 4.16T + 5T^{2} \)
7 \( 1 + 0.683T + 7T^{2} \)
13 \( 1 - 2.68T + 13T^{2} \)
17 \( 1 - 3.36T + 17T^{2} \)
23 \( 1 + 1.36T + 23T^{2} \)
29 \( 1 - 2.79T + 29T^{2} \)
31 \( 1 - 5.01T + 31T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 + 7.01T + 41T^{2} \)
43 \( 1 - 7.86T + 43T^{2} \)
47 \( 1 - 2.96T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 9.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 0.683T + 67T^{2} \)
71 \( 1 - 3.53T + 71T^{2} \)
73 \( 1 - 0.407T + 73T^{2} \)
79 \( 1 + 7.28T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−11.15318713195075482282247711275, −10.40866193169665718996320851585, −9.828055502274430804721902238363, −8.540511508341507227250707888885, −6.86228195712903422927850873640, −6.12474216890903363498684479580, −5.61935710735571675983611659647, −4.78676607391630482467669756030, −3.01107070329188969587230374623, −1.50771487045926067104229885805, 1.50771487045926067104229885805, 3.01107070329188969587230374623, 4.78676607391630482467669756030, 5.61935710735571675983611659647, 6.12474216890903363498684479580, 6.86228195712903422927850873640, 8.540511508341507227250707888885, 9.828055502274430804721902238363, 10.40866193169665718996320851585, 11.15318713195075482282247711275

Graph of the $Z$-function along the critical line