Properties

Degree $2$
Conductor $4704$
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 − 2·5-s + 9-s − 2·11-s − 2·15-s − 2·17-s + 2·23-s − 25-s + 27-s + 6·29-s + 4·31-s − 2·33-s + 6·37-s + 2·41-s − 2·45-s − 2·51-s − 6·53-s + 4·55-s − 12·59-s − 12·61-s − 12·67-s + 2·69-s − 10·71-s − 12·73-s − 75-s + 12·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s − 0.516·15-s − 0.485·17-s + 0.417·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.348·33-s + 0.986·37-s + 0.312·41-s − 0.298·45-s − 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.56·59-s − 1.53·61-s − 1.46·67-s + 0.240·69-s − 1.18·71-s − 1.40·73-s − 0.115·75-s + 1.35·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4704\)    =    \(2^{5} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{4704} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4704,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.87485564393543084588505050008, −7.48726070202434962945656021189, −6.58590651258021593485600290622, −5.81275009629452231890417758838, −4.60073105091703338282724299008, −4.34797922753245699749563191225, −3.16896073131960953058233034268, −2.67915576492671508720724712589, −1.39043549289335549525000689318, 0, 1.39043549289335549525000689318, 2.67915576492671508720724712589, 3.16896073131960953058233034268, 4.34797922753245699749563191225, 4.60073105091703338282724299008, 5.81275009629452231890417758838, 6.58590651258021593485600290622, 7.48726070202434962945656021189, 7.87485564393543084588505050008

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