Properties

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

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 7·5-s + 9·9-s − 7·11-s + 52·13-s + 21·15-s − 72·17-s + 20·19-s + 48·23-s − 76·25-s − 27·27-s − 243·29-s + 95·31-s + 21·33-s + 352·37-s − 156·39-s + 296·41-s − 158·43-s − 63·45-s − 142·47-s + 216·51-s − 375·53-s + 49·55-s − 60·57-s + 279·59-s − 246·61-s − 364·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.626·5-s + 1/3·9-s − 0.191·11-s + 1.10·13-s + 0.361·15-s − 1.02·17-s + 0.241·19-s + 0.435·23-s − 0.607·25-s − 0.192·27-s − 1.55·29-s + 0.550·31-s + 0.110·33-s + 1.56·37-s − 0.640·39-s + 1.12·41-s − 0.560·43-s − 0.208·45-s − 0.440·47-s + 0.593·51-s − 0.971·53-s + 0.120·55-s − 0.139·57-s + 0.615·59-s − 0.516·61-s − 0.694·65-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 \( 1 + 7 T + p^{3} T^{2} \)
11 \( 1 + 7 T + p^{3} T^{2} \)
13 \( 1 - 4 p T + p^{3} T^{2} \)
17 \( 1 + 72 T + p^{3} T^{2} \)
19 \( 1 - 20 T + p^{3} T^{2} \)
23 \( 1 - 48 T + p^{3} T^{2} \)
29 \( 1 + 243 T + p^{3} T^{2} \)
31 \( 1 - 95 T + p^{3} T^{2} \)
37 \( 1 - 352 T + p^{3} T^{2} \)
41 \( 1 - 296 T + p^{3} T^{2} \)
43 \( 1 + 158 T + p^{3} T^{2} \)
47 \( 1 + 142 T + p^{3} T^{2} \)
53 \( 1 + 375 T + p^{3} T^{2} \)
59 \( 1 - 279 T + p^{3} T^{2} \)
61 \( 1 + 246 T + p^{3} T^{2} \)
67 \( 1 - 730 T + p^{3} T^{2} \)
71 \( 1 + 338 T + p^{3} T^{2} \)
73 \( 1 - 542 T + p^{3} T^{2} \)
79 \( 1 - 305 T + p^{3} T^{2} \)
83 \( 1 - 1123 T + p^{3} T^{2} \)
89 \( 1 - 426 T + p^{3} T^{2} \)
97 \( 1 - 369 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.101428718066792016994711590806, −7.56941860436075316655534957980, −6.57197148611573938978276506770, −6.00688398564597290102268896787, −5.05722337916991833499952518612, −4.19188742393081378046953957537, −3.50866058074871512213941241847, −2.24644261707548758170075795747, −1.04394674450421689707309850836, 0, 1.04394674450421689707309850836, 2.24644261707548758170075795747, 3.50866058074871512213941241847, 4.19188742393081378046953957537, 5.05722337916991833499952518612, 6.00688398564597290102268896787, 6.57197148611573938978276506770, 7.56941860436075316655534957980, 8.101428718066792016994711590806

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