Properties

Label 2-11466-1.1-c1-0-13
Degree $2$
Conductor $11466$
Sign $1$
Analytic cond. $91.5564$
Root an. cond. $9.56851$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 11-s + 13-s + 16-s − 2·17-s + 2·19-s + 20-s + 22-s + 2·23-s − 4·25-s − 26-s + 7·29-s − 3·31-s − 32-s + 2·34-s − 2·37-s − 2·38-s − 40-s + 8·41-s − 8·43-s − 44-s − 2·46-s + 4·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.223·20-s + 0.213·22-s + 0.417·23-s − 4/5·25-s − 0.196·26-s + 1.29·29-s − 0.538·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.324·38-s − 0.158·40-s + 1.24·41-s − 1.21·43-s − 0.150·44-s − 0.294·46-s + 0.565·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11466\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(91.5564\)
Root analytic conductor: \(9.56851\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11466,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.485255647\)
\(L(\frac12)\) \(\approx\) \(1.485255647\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
good5 \( 1 - T + p T^{2} \) 1.5.ab
11 \( 1 + T + p T^{2} \) 1.11.b
17 \( 1 + 2 T + p T^{2} \) 1.17.c
19 \( 1 - 2 T + p T^{2} \) 1.19.ac
23 \( 1 - 2 T + p T^{2} \) 1.23.ac
29 \( 1 - 7 T + p T^{2} \) 1.29.ah
31 \( 1 + 3 T + p T^{2} \) 1.31.d
37 \( 1 + 2 T + p T^{2} \) 1.37.c
41 \( 1 - 8 T + p T^{2} \) 1.41.ai
43 \( 1 + 8 T + p T^{2} \) 1.43.i
47 \( 1 + p T^{2} \) 1.47.a
53 \( 1 - 3 T + p T^{2} \) 1.53.ad
59 \( 1 + 9 T + p T^{2} \) 1.59.j
61 \( 1 - 8 T + p T^{2} \) 1.61.ai
67 \( 1 + 2 T + p T^{2} \) 1.67.c
71 \( 1 - 16 T + p T^{2} \) 1.71.aq
73 \( 1 + 6 T + p T^{2} \) 1.73.g
79 \( 1 - 5 T + p T^{2} \) 1.79.af
83 \( 1 - 11 T + p T^{2} \) 1.83.al
89 \( 1 + 6 T + p T^{2} \) 1.89.g
97 \( 1 + T + p T^{2} \) 1.97.b
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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

−16.46788713588524, −15.89974658587799, −15.49963672640828, −14.83940352921363, −14.11515154722356, −13.63501996612062, −13.02732313961599, −12.37718134197768, −11.72874624063226, −11.15214630758491, −10.54950188811940, −10.03204483970442, −9.382865240541641, −8.914226837515496, −8.165374598319363, −7.688820691285508, −6.846300894267944, −6.385111037881819, −5.598364691625594, −4.982955985141765, −4.048677799601795, −3.173775758291352, −2.403076986657317, −1.624190808141456, −0.6319272388518665, 0.6319272388518665, 1.624190808141456, 2.403076986657317, 3.173775758291352, 4.048677799601795, 4.982955985141765, 5.598364691625594, 6.385111037881819, 6.846300894267944, 7.688820691285508, 8.165374598319363, 8.914226837515496, 9.382865240541641, 10.03204483970442, 10.54950188811940, 11.15214630758491, 11.72874624063226, 12.37718134197768, 13.02732313961599, 13.63501996612062, 14.11515154722356, 14.83940352921363, 15.49963672640828, 15.89974658587799, 16.46788713588524

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