Properties

Label 2-322-1.1-c3-0-26
Degree $2$
Conductor $322$
Sign $-1$
Analytic cond. $18.9986$
Root an. cond. $4.35874$
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 + 6.22·3-s + 4·4-s − 9.89·5-s − 12.4·6-s + 7·7-s − 8·8-s + 11.7·9-s + 19.7·10-s − 41.5·11-s + 24.9·12-s + 18.7·13-s − 14·14-s − 61.5·15-s + 16·16-s − 19.3·17-s − 23.5·18-s − 52.3·19-s − 39.5·20-s + 43.5·21-s + 83.0·22-s − 23·23-s − 49.8·24-s − 27.1·25-s − 37.4·26-s − 94.8·27-s + 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.19·3-s + 0.5·4-s − 0.884·5-s − 0.847·6-s + 0.377·7-s − 0.353·8-s + 0.435·9-s + 0.625·10-s − 1.13·11-s + 0.599·12-s + 0.399·13-s − 0.267·14-s − 1.05·15-s + 0.250·16-s − 0.275·17-s − 0.308·18-s − 0.631·19-s − 0.442·20-s + 0.452·21-s + 0.804·22-s − 0.208·23-s − 0.423·24-s − 0.217·25-s − 0.282·26-s − 0.676·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(18.9986\)
Root analytic conductor: \(4.35874\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 322,\ (\ :3/2),\ -1)\)

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 \( 1 + 2T \)
7 \( 1 - 7T \)
23 \( 1 + 23T \)
good3 \( 1 - 6.22T + 27T^{2} \)
5 \( 1 + 9.89T + 125T^{2} \)
11 \( 1 + 41.5T + 1.33e3T^{2} \)
13 \( 1 - 18.7T + 2.19e3T^{2} \)
17 \( 1 + 19.3T + 4.91e3T^{2} \)
19 \( 1 + 52.3T + 6.85e3T^{2} \)
29 \( 1 + 169.T + 2.43e4T^{2} \)
31 \( 1 + 191.T + 2.97e4T^{2} \)
37 \( 1 - 366.T + 5.06e4T^{2} \)
41 \( 1 - 81.0T + 6.89e4T^{2} \)
43 \( 1 - 202.T + 7.95e4T^{2} \)
47 \( 1 - 86.9T + 1.03e5T^{2} \)
53 \( 1 + 708.T + 1.48e5T^{2} \)
59 \( 1 + 618.T + 2.05e5T^{2} \)
61 \( 1 + 258.T + 2.26e5T^{2} \)
67 \( 1 + 167.T + 3.00e5T^{2} \)
71 \( 1 + 281.T + 3.57e5T^{2} \)
73 \( 1 + 643.T + 3.89e5T^{2} \)
79 \( 1 + 397.T + 4.93e5T^{2} \)
83 \( 1 - 1.43e3T + 5.71e5T^{2} \)
89 \( 1 + 497.T + 7.04e5T^{2} \)
97 \( 1 - 193.T + 9.12e5T^{2} \)
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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

−10.78215733107329165531589588682, −9.506249200061919314125822343294, −8.739715334958647901547510108959, −7.78756122622661435685464281531, −7.61769800177540783422810974398, −5.91269618559784720465206250691, −4.29108360984605284501629230687, −3.12900740479843562859264024369, −1.98895476068483418595305245526, 0, 1.98895476068483418595305245526, 3.12900740479843562859264024369, 4.29108360984605284501629230687, 5.91269618559784720465206250691, 7.61769800177540783422810974398, 7.78756122622661435685464281531, 8.739715334958647901547510108959, 9.506249200061919314125822343294, 10.78215733107329165531589588682

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