Properties

Degree $2$
Conductor $13328$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 6·11-s − 2·13-s + 17-s − 4·19-s − 5·25-s + 4·27-s − 4·31-s + 12·33-s − 4·37-s + 4·39-s − 6·41-s − 8·43-s − 2·51-s − 6·53-s + 8·57-s + 4·61-s − 8·67-s − 2·73-s + 10·75-s − 8·79-s − 11·81-s + 6·89-s + 8·93-s − 14·97-s − 6·99-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 25-s + 0.769·27-s − 0.718·31-s + 2.08·33-s − 0.657·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.280·51-s − 0.824·53-s + 1.05·57-s + 0.512·61-s − 0.977·67-s − 0.234·73-s + 1.15·75-s − 0.900·79-s − 1.22·81-s + 0.635·89-s + 0.829·93-s − 1.42·97-s − 0.603·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13328\)    =    \(2^{4} \cdot 7^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{13328} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 13328,\ (\ :1/2),\ 1)\)

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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.89938879345945, −16.10399920112528, −15.95557330708872, −15.06193438769964, −14.79362371269530, −13.79950559221862, −13.33031016271785, −12.74623150851431, −12.19678466193077, −11.73800526495891, −10.95408261021705, −10.64027386824301, −10.08065348148141, −9.512907554292647, −8.443655717530582, −8.119088937246102, −7.300319633094969, −6.742109358879015, −5.950598949559237, −5.393194043090607, −5.021222709598197, −4.253836594933654, −3.235128551131609, −2.471657038647939, −1.593493697187460, 0, 0, 1.593493697187460, 2.471657038647939, 3.235128551131609, 4.253836594933654, 5.021222709598197, 5.393194043090607, 5.950598949559237, 6.742109358879015, 7.300319633094969, 8.119088937246102, 8.443655717530582, 9.512907554292647, 10.08065348148141, 10.64027386824301, 10.95408261021705, 11.73800526495891, 12.19678466193077, 12.74623150851431, 13.33031016271785, 13.79950559221862, 14.79362371269530, 15.06193438769964, 15.95557330708872, 16.10399920112528, 16.89938879345945

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