Properties

Label 2-414-1.1-c3-0-21
Degree $2$
Conductor $414$
Sign $-1$
Analytic cond. $24.4267$
Root an. cond. $4.94234$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 7.31·5-s + 17.3·7-s − 8·8-s − 14.6·10-s − 69.2·11-s − 3.25·13-s − 34.6·14-s + 16·16-s − 85.9·17-s − 19.9·19-s + 29.2·20-s + 138.·22-s − 23·23-s − 71.5·25-s + 6.50·26-s + 69.2·28-s + 47.1·29-s + 109.·31-s − 32·32-s + 171.·34-s + 126.·35-s + 86.9·37-s + 39.8·38-s − 58.5·40-s − 65.7·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.654·5-s + 0.934·7-s − 0.353·8-s − 0.462·10-s − 1.89·11-s − 0.0694·13-s − 0.661·14-s + 0.250·16-s − 1.22·17-s − 0.240·19-s + 0.327·20-s + 1.34·22-s − 0.208·23-s − 0.572·25-s + 0.0491·26-s + 0.467·28-s + 0.301·29-s + 0.632·31-s − 0.176·32-s + 0.866·34-s + 0.611·35-s + 0.386·37-s + 0.170·38-s − 0.231·40-s − 0.250·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(24.4267\)
Root analytic conductor: \(4.94234\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 414,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 \)
23 \( 1 + 23T \)
good5 \( 1 - 7.31T + 125T^{2} \)
7 \( 1 - 17.3T + 343T^{2} \)
11 \( 1 + 69.2T + 1.33e3T^{2} \)
13 \( 1 + 3.25T + 2.19e3T^{2} \)
17 \( 1 + 85.9T + 4.91e3T^{2} \)
19 \( 1 + 19.9T + 6.85e3T^{2} \)
29 \( 1 - 47.1T + 2.43e4T^{2} \)
31 \( 1 - 109.T + 2.97e4T^{2} \)
37 \( 1 - 86.9T + 5.06e4T^{2} \)
41 \( 1 + 65.7T + 6.89e4T^{2} \)
43 \( 1 + 398.T + 7.95e4T^{2} \)
47 \( 1 - 164.T + 1.03e5T^{2} \)
53 \( 1 + 631.T + 1.48e5T^{2} \)
59 \( 1 - 665.T + 2.05e5T^{2} \)
61 \( 1 + 490.T + 2.26e5T^{2} \)
67 \( 1 - 83.4T + 3.00e5T^{2} \)
71 \( 1 + 969.T + 3.57e5T^{2} \)
73 \( 1 - 462.T + 3.89e5T^{2} \)
79 \( 1 + 857.T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3T + 5.71e5T^{2} \)
89 \( 1 - 382.T + 7.04e5T^{2} \)
97 \( 1 + 817.T + 9.12e5T^{2} \)
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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

−10.38736563868443191599140933450, −9.504800213098410368870686183925, −8.348867092084888479250315466615, −7.87726253472385184918461047146, −6.67091035029967032304593447353, −5.54454589810460713650879819794, −4.62503409436491508097606741070, −2.69751051312056910549723364037, −1.79049007682542678604872165044, 0, 1.79049007682542678604872165044, 2.69751051312056910549723364037, 4.62503409436491508097606741070, 5.54454589810460713650879819794, 6.67091035029967032304593447353, 7.87726253472385184918461047146, 8.348867092084888479250315466615, 9.504800213098410368870686183925, 10.38736563868443191599140933450

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