Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7^{2} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 18·5-s + 9·9-s + 36·11-s + 34·13-s − 54·15-s − 42·17-s − 124·19-s + 199·25-s − 27·27-s + 102·29-s − 160·31-s − 108·33-s + 398·37-s − 102·39-s + 318·41-s + 268·43-s + 162·45-s + 240·47-s + 126·51-s − 498·53-s + 648·55-s + 372·57-s − 132·59-s − 398·61-s + 612·65-s − 92·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.60·5-s + 1/3·9-s + 0.986·11-s + 0.725·13-s − 0.929·15-s − 0.599·17-s − 1.49·19-s + 1.59·25-s − 0.192·27-s + 0.653·29-s − 0.926·31-s − 0.569·33-s + 1.76·37-s − 0.418·39-s + 1.21·41-s + 0.950·43-s + 0.536·45-s + 0.744·47-s + 0.345·51-s − 1.29·53-s + 1.58·55-s + 0.864·57-s − 0.291·59-s − 0.835·61-s + 1.16·65-s − 0.167·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{2352} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2352,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(2.916233193\)
\(L(\frac12)\)  \(\approx\)  \(2.916233193\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p T \)
7 \( 1 \)
good5 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 - 34 T + p^{3} T^{2} \)
17 \( 1 + 42 T + p^{3} T^{2} \)
19 \( 1 + 124 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 - 102 T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 - 398 T + p^{3} T^{2} \)
41 \( 1 - 318 T + p^{3} T^{2} \)
43 \( 1 - 268 T + p^{3} T^{2} \)
47 \( 1 - 240 T + p^{3} T^{2} \)
53 \( 1 + 498 T + p^{3} T^{2} \)
59 \( 1 + 132 T + p^{3} T^{2} \)
61 \( 1 + 398 T + p^{3} T^{2} \)
67 \( 1 + 92 T + p^{3} T^{2} \)
71 \( 1 - 720 T + p^{3} T^{2} \)
73 \( 1 - 502 T + p^{3} T^{2} \)
79 \( 1 - 1024 T + p^{3} T^{2} \)
83 \( 1 + 204 T + p^{3} T^{2} \)
89 \( 1 + 354 T + p^{3} T^{2} \)
97 \( 1 - 286 T + p^{3} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.019508717803575527744061221169, −7.86547854739755706506626502646, −6.65780994245299799780047031079, −6.25899151641670872735894815522, −5.78167064393678698914988657987, −4.68981509002075214534536436060, −3.95960720729183619520530011459, −2.53980161648725915503270326574, −1.74879218677032703642389102665, −0.816630057123859507244975008260, 0.816630057123859507244975008260, 1.74879218677032703642389102665, 2.53980161648725915503270326574, 3.95960720729183619520530011459, 4.68981509002075214534536436060, 5.78167064393678698914988657987, 6.25899151641670872735894815522, 6.65780994245299799780047031079, 7.86547854739755706506626502646, 9.019508717803575527744061221169

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