Properties

Label 2-22-1.1-c9-0-2
Degree $2$
Conductor $22$
Sign $1$
Analytic cond. $11.3307$
Root an. cond. $3.36612$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 137·3-s + 256·4-s − 595·5-s + 2.19e3·6-s + 1.13e4·7-s + 4.09e3·8-s − 914·9-s − 9.52e3·10-s − 1.46e4·11-s + 3.50e4·12-s + 5.56e4·13-s + 1.81e5·14-s − 8.15e4·15-s + 6.55e4·16-s + 4.21e5·17-s − 1.46e4·18-s − 4.35e5·19-s − 1.52e5·20-s + 1.55e6·21-s − 2.34e5·22-s + 7.79e5·23-s + 5.61e5·24-s − 1.59e6·25-s + 8.89e5·26-s − 2.82e6·27-s + 2.90e6·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.976·3-s + 1/2·4-s − 0.425·5-s + 0.690·6-s + 1.78·7-s + 0.353·8-s − 0.0464·9-s − 0.301·10-s − 0.301·11-s + 0.488·12-s + 0.540·13-s + 1.26·14-s − 0.415·15-s + 1/4·16-s + 1.22·17-s − 0.0328·18-s − 0.767·19-s − 0.212·20-s + 1.74·21-s − 0.213·22-s + 0.580·23-s + 0.345·24-s − 0.818·25-s + 0.381·26-s − 1.02·27-s + 0.893·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $1$
Analytic conductor: \(11.3307\)
Root analytic conductor: \(3.36612\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(3.777404390\)
\(L(\frac12)\) \(\approx\) \(3.777404390\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{4} T \)
11 \( 1 + p^{4} T \)
good3 \( 1 - 137 T + p^{9} T^{2} \)
5 \( 1 + 119 p T + p^{9} T^{2} \)
7 \( 1 - 1622 p T + p^{9} T^{2} \)
13 \( 1 - 55620 T + p^{9} T^{2} \)
17 \( 1 - 421550 T + p^{9} T^{2} \)
19 \( 1 + 435872 T + p^{9} T^{2} \)
23 \( 1 - 779077 T + p^{9} T^{2} \)
29 \( 1 - 1206768 T + p^{9} T^{2} \)
31 \( 1 + 7626195 T + p^{9} T^{2} \)
37 \( 1 + 19473681 T + p^{9} T^{2} \)
41 \( 1 + 9906168 T + p^{9} T^{2} \)
43 \( 1 + 20662730 T + p^{9} T^{2} \)
47 \( 1 - 2751600 T + p^{9} T^{2} \)
53 \( 1 - 78527114 T + p^{9} T^{2} \)
59 \( 1 + 105727893 T + p^{9} T^{2} \)
61 \( 1 + 24639860 T + p^{9} T^{2} \)
67 \( 1 + 94817369 T + p^{9} T^{2} \)
71 \( 1 - 338924343 T + p^{9} T^{2} \)
73 \( 1 - 10287972 T + p^{9} T^{2} \)
79 \( 1 - 556386626 T + p^{9} T^{2} \)
83 \( 1 - 22479190 T + p^{9} T^{2} \)
89 \( 1 + 213504377 T + p^{9} T^{2} \)
97 \( 1 - 1512644953 T + p^{9} T^{2} \)
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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

−15.30048706081349747751014004302, −14.56286022240412711029392365441, −13.66224107692048883566376965352, −11.98491091928900548819855946054, −10.81426654216504142344227045614, −8.562579199117801393742967285805, −7.64685484719148717564965556224, −5.25961820130201723537348793567, −3.63057057222498432959683611351, −1.84086124668774398099564633120, 1.84086124668774398099564633120, 3.63057057222498432959683611351, 5.25961820130201723537348793567, 7.64685484719148717564965556224, 8.562579199117801393742967285805, 10.81426654216504142344227045614, 11.98491091928900548819855946054, 13.66224107692048883566376965352, 14.56286022240412711029392365441, 15.30048706081349747751014004302

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