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-s − 2·9-s + 5·11-s − 5·13-s + 17-s − 6·19-s + 4·23-s − 5·25-s + 5·27-s − 4·29-s − 5·33-s − 8·37-s + 5·39-s − 4·41-s + 6·43-s + 6·47-s − 51-s − 11·53-s + 6·57-s + 10·59-s − 10·67-s − 4·69-s + 9·71-s − 4·73-s + 5·75-s + 9·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s + 1.50·11-s − 1.38·13-s + 0.242·17-s − 1.37·19-s + 0.834·23-s − 25-s + 0.962·27-s − 0.742·29-s − 0.870·33-s − 1.31·37-s + 0.800·39-s − 0.624·41-s + 0.914·43-s + 0.875·47-s − 0.140·51-s − 1.51·53-s + 0.794·57-s + 1.30·59-s − 1.22·67-s − 0.481·69-s + 1.06·71-s − 0.468·73-s + 0.577·75-s + 1.01·79-s + 1/9·81-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 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−14.66678110861818, −14.23732438296921, −13.88263976458038, −12.99515113594252, −12.59219088513227, −12.04579511804924, −11.66788988500352, −11.29298572887756, −10.50245150029557, −10.27054841168261, −9.359895031017861, −9.089434193310411, −8.604405209241406, −7.768734208147357, −7.332554467398347, −6.555523476635233, −6.352150748313330, −5.608718558138234, −5.053180520443816, −4.508348141990801, −3.799556596244893, −3.253389653972057, −2.300830404095696, −1.836477007571271, −0.7828795126668720, 0, 0.7828795126668720, 1.836477007571271, 2.300830404095696, 3.253389653972057, 3.799556596244893, 4.508348141990801, 5.053180520443816, 5.608718558138234, 6.352150748313330, 6.555523476635233, 7.332554467398347, 7.768734208147357, 8.604405209241406, 9.089434193310411, 9.359895031017861, 10.27054841168261, 10.50245150029557, 11.29298572887756, 11.66788988500352, 12.04579511804924, 12.59219088513227, 12.99515113594252, 13.88263976458038, 14.23732438296921, 14.66678110861818

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