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  − 2·5-s − 3·9-s − 2·13-s − 17-s − 4·19-s + 4·23-s − 25-s − 6·29-s − 4·31-s + 2·37-s + 6·41-s − 4·43-s + 6·45-s − 6·53-s − 12·59-s − 10·61-s + 4·65-s − 4·67-s − 4·71-s + 6·73-s + 12·79-s + 9·81-s − 4·83-s + 2·85-s − 10·89-s + 8·95-s − 2·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s − 0.554·13-s − 0.242·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.894·45-s − 0.824·53-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 0.488·67-s − 0.474·71-s + 0.702·73-s + 1.35·79-s + 81-s − 0.439·83-s + 0.216·85-s − 1.05·89-s + 0.820·95-s − 0.203·97-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 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.98376818011324, −14.67652030345076, −13.90025087775239, −13.50291622937111, −12.78603715547769, −12.39385392131390, −11.92046132854287, −11.25642480802247, −10.96646977280435, −10.60706999965465, −9.582780906964060, −9.264229860297618, −8.744638842722893, −8.005443599350900, −7.788655604463327, −7.126079170619506, −6.502631621726154, −5.900790565010751, −5.322374664086590, −4.657501630557918, −4.117945284481163, −3.460754549714821, −2.862762107733931, −2.212861659701392, −1.346507847854587, 0, 0, 1.346507847854587, 2.212861659701392, 2.862762107733931, 3.460754549714821, 4.117945284481163, 4.657501630557918, 5.322374664086590, 5.900790565010751, 6.502631621726154, 7.126079170619506, 7.788655604463327, 8.005443599350900, 8.744638842722893, 9.264229860297618, 9.582780906964060, 10.60706999965465, 10.96646977280435, 11.25642480802247, 11.92046132854287, 12.39385392131390, 12.78603715547769, 13.50291622937111, 13.90025087775239, 14.67652030345076, 14.98376818011324

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