Properties

Label 2-22-1.1-c3-0-0
Degree $2$
Conductor $22$
Sign $1$
Analytic cond. $1.29804$
Root an. cond. $1.13931$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·3-s + 4·4-s + 14·5-s − 8·6-s − 8·7-s − 8·8-s − 11·9-s − 28·10-s − 11·11-s + 16·12-s − 50·13-s + 16·14-s + 56·15-s + 16·16-s + 130·17-s + 22·18-s − 108·19-s + 56·20-s − 32·21-s + 22·22-s − 96·23-s − 32·24-s + 71·25-s + 100·26-s − 152·27-s − 32·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.769·3-s + 1/2·4-s + 1.25·5-s − 0.544·6-s − 0.431·7-s − 0.353·8-s − 0.407·9-s − 0.885·10-s − 0.301·11-s + 0.384·12-s − 1.06·13-s + 0.305·14-s + 0.963·15-s + 1/4·16-s + 1.85·17-s + 0.288·18-s − 1.30·19-s + 0.626·20-s − 0.332·21-s + 0.213·22-s − 0.870·23-s − 0.272·24-s + 0.567·25-s + 0.754·26-s − 1.08·27-s − 0.215·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $1$
Analytic conductor: \(1.29804\)
Root analytic conductor: \(1.13931\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.077378283\)
\(L(\frac12)\) \(\approx\) \(1.077378283\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
11 \( 1 + p T \)
good3 \( 1 - 4 T + p^{3} T^{2} \)
5 \( 1 - 14 T + p^{3} T^{2} \)
7 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 + 50 T + p^{3} T^{2} \)
17 \( 1 - 130 T + p^{3} T^{2} \)
19 \( 1 + 108 T + p^{3} T^{2} \)
23 \( 1 + 96 T + p^{3} T^{2} \)
29 \( 1 - 142 T + p^{3} T^{2} \)
31 \( 1 - 40 T + p^{3} T^{2} \)
37 \( 1 - 382 T + p^{3} T^{2} \)
41 \( 1 + 118 T + p^{3} T^{2} \)
43 \( 1 - 220 T + p^{3} T^{2} \)
47 \( 1 - 520 T + p^{3} T^{2} \)
53 \( 1 - 238 T + p^{3} T^{2} \)
59 \( 1 + 852 T + p^{3} T^{2} \)
61 \( 1 - 190 T + p^{3} T^{2} \)
67 \( 1 + 12 T + p^{3} T^{2} \)
71 \( 1 + 112 T + p^{3} T^{2} \)
73 \( 1 + 6 T + p^{3} T^{2} \)
79 \( 1 - 304 T + p^{3} T^{2} \)
83 \( 1 - 820 T + p^{3} T^{2} \)
89 \( 1 - 202 T + p^{3} T^{2} \)
97 \( 1 + 1406 T + p^{3} T^{2} \)
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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

−17.47494161568956176720036893642, −16.62458391905852663211802066251, −14.85775313266196954889461677486, −13.88874549226350195826085934195, −12.37329896898312354317657219907, −10.25733214586728602648713961547, −9.414938067932582923275046319275, −7.926629231949739540911245680424, −5.96419217544703509793971274784, −2.53420288893902314690330600827, 2.53420288893902314690330600827, 5.96419217544703509793971274784, 7.926629231949739540911245680424, 9.414938067932582923275046319275, 10.25733214586728602648713961547, 12.37329896898312354317657219907, 13.88874549226350195826085934195, 14.85775313266196954889461677486, 16.62458391905852663211802066251, 17.47494161568956176720036893642

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