Properties

Degree $2$
Conductor $53312$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s − 2·9-s − 3·11-s − 5·13-s + 4·15-s + 17-s − 6·19-s + 4·23-s + 11·25-s + 5·27-s − 8·29-s + 3·33-s + 8·37-s + 5·39-s + 8·41-s + 10·43-s + 8·45-s − 2·47-s − 51-s − 3·53-s + 12·55-s + 6·57-s − 2·59-s − 8·61-s + 20·65-s + 14·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 2/3·9-s − 0.904·11-s − 1.38·13-s + 1.03·15-s + 0.242·17-s − 1.37·19-s + 0.834·23-s + 11/5·25-s + 0.962·27-s − 1.48·29-s + 0.522·33-s + 1.31·37-s + 0.800·39-s + 1.24·41-s + 1.52·43-s + 1.19·45-s − 0.291·47-s − 0.140·51-s − 0.412·53-s + 1.61·55-s + 0.794·57-s − 0.260·59-s − 1.02·61-s + 2.48·65-s + 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53312\)    =    \(2^{6} \cdot 7^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{53312} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.98315409147031, −14.56078423670992, −14.23101227360606, −13.11263989619804, −12.67225757714595, −12.51375290740198, −11.86039971054657, −11.23488137318044, −11.03210899851035, −10.66617991779924, −9.812890306345039, −9.215445116417312, −8.610176929211600, −8.009622777933602, −7.639030965605177, −7.225805870205977, −6.578635597773967, −5.768124240941405, −5.342411933902495, −4.489025600816516, −4.393206195874410, −3.511267501004365, −2.726251095400751, −2.427864208599047, −1.017518343482637, 0, 0, 1.017518343482637, 2.427864208599047, 2.726251095400751, 3.511267501004365, 4.393206195874410, 4.489025600816516, 5.342411933902495, 5.768124240941405, 6.578635597773967, 7.225805870205977, 7.639030965605177, 8.009622777933602, 8.610176929211600, 9.215445116417312, 9.812890306345039, 10.66617991779924, 11.03210899851035, 11.23488137318044, 11.86039971054657, 12.51375290740198, 12.67225757714595, 13.11263989619804, 14.23101227360606, 14.56078423670992, 14.98315409147031

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