Properties

Degree $2$
Conductor $2352$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 3·5-s + 9·9-s + 15·11-s + 64·13-s + 9·15-s − 84·17-s − 16·19-s + 84·23-s − 116·25-s + 27·27-s − 297·29-s − 253·31-s + 45·33-s − 316·37-s + 192·39-s − 360·41-s − 26·43-s + 27·45-s − 30·47-s − 252·51-s + 363·53-s + 45·55-s − 48·57-s − 15·59-s + 118·61-s + 192·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.268·5-s + 1/3·9-s + 0.411·11-s + 1.36·13-s + 0.154·15-s − 1.19·17-s − 0.193·19-s + 0.761·23-s − 0.927·25-s + 0.192·27-s − 1.90·29-s − 1.46·31-s + 0.237·33-s − 1.40·37-s + 0.788·39-s − 1.37·41-s − 0.0922·43-s + 0.0894·45-s − 0.0931·47-s − 0.691·51-s + 0.940·53-s + 0.110·55-s − 0.111·57-s − 0.0330·59-s + 0.247·61-s + 0.366·65-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

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

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
7 \( 1 \)
good5 \( 1 - 3 T + p^{3} T^{2} \)
11 \( 1 - 15 T + p^{3} T^{2} \)
13 \( 1 - 64 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 + 16 T + p^{3} T^{2} \)
23 \( 1 - 84 T + p^{3} T^{2} \)
29 \( 1 + 297 T + p^{3} T^{2} \)
31 \( 1 + 253 T + p^{3} T^{2} \)
37 \( 1 + 316 T + p^{3} T^{2} \)
41 \( 1 + 360 T + p^{3} T^{2} \)
43 \( 1 + 26 T + p^{3} T^{2} \)
47 \( 1 + 30 T + p^{3} T^{2} \)
53 \( 1 - 363 T + p^{3} T^{2} \)
59 \( 1 + 15 T + p^{3} T^{2} \)
61 \( 1 - 118 T + p^{3} T^{2} \)
67 \( 1 - 370 T + p^{3} T^{2} \)
71 \( 1 - 342 T + p^{3} T^{2} \)
73 \( 1 + 362 T + p^{3} T^{2} \)
79 \( 1 + 467 T + p^{3} T^{2} \)
83 \( 1 - 477 T + p^{3} T^{2} \)
89 \( 1 + 906 T + p^{3} T^{2} \)
97 \( 1 + 503 T + p^{3} T^{2} \)
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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

−8.538106442656313879908579130193, −7.41945651372202687786155297726, −6.79221194929837980216803206923, −5.93777594421574156551358414107, −5.11358253552231939908642827230, −3.88716262973312236521405538879, −3.52940137605478574098691127308, −2.14220364139252212553734794780, −1.50692040480369034830262916231, 0, 1.50692040480369034830262916231, 2.14220364139252212553734794780, 3.52940137605478574098691127308, 3.88716262973312236521405538879, 5.11358253552231939908642827230, 5.93777594421574156551358414107, 6.79221194929837980216803206923, 7.41945651372202687786155297726, 8.538106442656313879908579130193

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