Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·5-s + 6·9-s − 5·11-s − 7·13-s − 6·15-s + 17-s + 2·19-s − 25-s − 9·27-s − 6·29-s + 4·31-s + 15·33-s + 8·37-s + 21·39-s − 2·41-s − 8·43-s + 12·45-s − 10·47-s − 3·51-s + 3·53-s − 10·55-s − 6·57-s + 12·61-s − 14·65-s − 2·67-s + 71-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.894·5-s + 2·9-s − 1.50·11-s − 1.94·13-s − 1.54·15-s + 0.242·17-s + 0.458·19-s − 1/5·25-s − 1.73·27-s − 1.11·29-s + 0.718·31-s + 2.61·33-s + 1.31·37-s + 3.36·39-s − 0.312·41-s − 1.21·43-s + 1.78·45-s − 1.45·47-s − 0.420·51-s + 0.412·53-s − 1.34·55-s − 0.794·57-s + 1.53·61-s − 1.73·65-s − 0.244·67-s + 0.118·71-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: \(1\)
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 \( 1 + p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.90693002037730, −14.15512766665973, −13.42326018600698, −13.02464011200494, −12.75730631562612, −11.93306670144574, −11.79348427984423, −11.12514740857941, −10.54521252692900, −10.07197275912590, −9.750099728572547, −9.443566154161585, −8.242808192136098, −7.727835542489029, −7.251201482393660, −6.636694826422900, −6.106275877905619, −5.468845146562127, −5.062986234600519, −4.945625024441987, −4.025758747371650, −2.983376990762342, −2.321590907095338, −1.728887311428013, −0.6715797427515093, 0, 0.6715797427515093, 1.728887311428013, 2.321590907095338, 2.983376990762342, 4.025758747371650, 4.945625024441987, 5.062986234600519, 5.468845146562127, 6.106275877905619, 6.636694826422900, 7.251201482393660, 7.727835542489029, 8.242808192136098, 9.443566154161585, 9.750099728572547, 10.07197275912590, 10.54521252692900, 11.12514740857941, 11.79348427984423, 11.93306670144574, 12.75730631562612, 13.02464011200494, 13.42326018600698, 14.15512766665973, 14.90693002037730

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