Properties

Label 2-11466-1.1-c1-0-17
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 − 8-s + 5·11-s − 13-s + 16-s − 7·17-s + 7·19-s − 5·22-s − 2·23-s − 5·25-s + 26-s + 9·29-s − 32-s + 7·34-s + 4·37-s − 7·38-s − 4·41-s + 2·43-s + 5·44-s + 2·46-s + 3·47-s + 5·50-s − 52-s − 53-s − 9·58-s − 7·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.50·11-s − 0.277·13-s + 1/4·16-s − 1.69·17-s + 1.60·19-s − 1.06·22-s − 0.417·23-s − 25-s + 0.196·26-s + 1.67·29-s − 0.176·32-s + 1.20·34-s + 0.657·37-s − 1.13·38-s − 0.624·41-s + 0.304·43-s + 0.753·44-s + 0.294·46-s + 0.437·47-s + 0.707·50-s − 0.138·52-s − 0.137·53-s − 1.18·58-s − 0.911·59-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

\(L(1)\) \(\approx\) \(1.542776944\)
\(L(\frac12)\) \(\approx\) \(1.542776944\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \) 1.5.a
11 \( 1 - 5 T + p T^{2} \) 1.11.af
17 \( 1 + 7 T + p T^{2} \) 1.17.h
19 \( 1 - 7 T + p T^{2} \) 1.19.ah
23 \( 1 + 2 T + p T^{2} \) 1.23.c
29 \( 1 - 9 T + p T^{2} \) 1.29.aj
31 \( 1 + p T^{2} \) 1.31.a
37 \( 1 - 4 T + p T^{2} \) 1.37.ae
41 \( 1 + 4 T + p T^{2} \) 1.41.e
43 \( 1 - 2 T + p T^{2} \) 1.43.ac
47 \( 1 - 3 T + p T^{2} \) 1.47.ad
53 \( 1 + T + p T^{2} \) 1.53.b
59 \( 1 + 7 T + p T^{2} \) 1.59.h
61 \( 1 - 13 T + p T^{2} \) 1.61.an
67 \( 1 - 3 T + p T^{2} \) 1.67.ad
71 \( 1 + 9 T + p T^{2} \) 1.71.j
73 \( 1 + 10 T + p T^{2} \) 1.73.k
79 \( 1 - 14 T + p T^{2} \) 1.79.ao
83 \( 1 + 16 T + p T^{2} \) 1.83.q
89 \( 1 - 12 T + p T^{2} \) 1.89.am
97 \( 1 - 6 T + p T^{2} \) 1.97.ag
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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

−16.38712123942376, −15.94671091021222, −15.49436484190468, −14.80815550145505, −14.09566354272495, −13.77374792992629, −13.02231274060969, −12.13285807746577, −11.72536207464342, −11.40036561133666, −10.56669366176041, −9.881276492133895, −9.429051639694688, −8.871083905111057, −8.303305564115244, −7.521630797588817, −6.908660085688210, −6.391167288140155, −5.747977535693094, −4.725309934880064, −4.121586643050782, −3.269991836481698, −2.387438951432516, −1.556880874123110, −0.6626860901384381, 0.6626860901384381, 1.556880874123110, 2.387438951432516, 3.269991836481698, 4.121586643050782, 4.725309934880064, 5.747977535693094, 6.391167288140155, 6.908660085688210, 7.521630797588817, 8.303305564115244, 8.871083905111057, 9.429051639694688, 9.881276492133895, 10.56669366176041, 11.40036561133666, 11.72536207464342, 12.13285807746577, 13.02231274060969, 13.77374792992629, 14.09566354272495, 14.80815550145505, 15.49436484190468, 15.94671091021222, 16.38712123942376

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