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

−7.900200107036443557888632457578, −7.19565425199867604846984703457, −6.49144949422330064889655352757, −5.84176384791580753826316565868, −4.85860065633960213694957415344, −4.20350188039766470578075947561, −3.55650103183881699712452674280, −2.40251776418476467417549003132, −1.17821699530441005692002962999, 0, 1.17821699530441005692002962999, 2.40251776418476467417549003132, 3.55650103183881699712452674280, 4.20350188039766470578075947561, 4.85860065633960213694957415344, 5.84176384791580753826316565868, 6.49144949422330064889655352757, 7.19565425199867604846984703457, 7.900200107036443557888632457578

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