Properties

Degree $2$
Conductor $2352$
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 − 4·13-s − 2·15-s + 6·17-s − 8·19-s + 6·23-s − 25-s − 27-s − 10·29-s − 4·31-s + 2·33-s + 6·37-s + 4·39-s − 6·41-s − 4·43-s + 2·45-s − 8·47-s − 6·51-s + 2·53-s − 4·55-s + 8·57-s + 4·59-s − 8·61-s − 8·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.516·15-s + 1.45·17-s − 1.83·19-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.348·33-s + 0.986·37-s + 0.640·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s − 0.840·51-s + 0.274·53-s − 0.539·55-s + 1.05·57-s + 0.520·59-s − 1.02·61-s − 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -1)\)

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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.631099201560386317062119026253, −7.71592026516063918650723966264, −7.02288599719673663589831070636, −6.14084677712433387704936155208, −5.41526796219077428009526667120, −4.90441675593865689533188061580, −3.70870379307642839783383641799, −2.52498891624783975371956159355, −1.63174134659533958289094579310, 0, 1.63174134659533958289094579310, 2.52498891624783975371956159355, 3.70870379307642839783383641799, 4.90441675593865689533188061580, 5.41526796219077428009526667120, 6.14084677712433387704936155208, 7.02288599719673663589831070636, 7.71592026516063918650723966264, 8.631099201560386317062119026253

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