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 + 18·5-s + 9·9-s − 36·11-s + 10·13-s + 54·15-s − 18·17-s − 100·19-s − 72·23-s + 199·25-s + 27·27-s − 234·29-s − 16·31-s − 108·33-s − 226·37-s + 30·39-s − 90·41-s − 452·43-s + 162·45-s + 432·47-s − 54·51-s + 414·53-s − 648·55-s − 300·57-s − 684·59-s − 422·61-s + 180·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.60·5-s + 1/3·9-s − 0.986·11-s + 0.213·13-s + 0.929·15-s − 0.256·17-s − 1.20·19-s − 0.652·23-s + 1.59·25-s + 0.192·27-s − 1.49·29-s − 0.0926·31-s − 0.569·33-s − 1.00·37-s + 0.123·39-s − 0.342·41-s − 1.60·43-s + 0.536·45-s + 1.34·47-s − 0.148·51-s + 1.07·53-s − 1.58·55-s − 0.697·57-s − 1.50·59-s − 0.885·61-s + 0.343·65-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 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 + 36 T + p^{3} T^{2} \)
13 \( 1 - 10 T + p^{3} T^{2} \)
17 \( 1 + 18 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 + 72 T + p^{3} T^{2} \)
29 \( 1 + 234 T + p^{3} T^{2} \)
31 \( 1 + 16 T + p^{3} T^{2} \)
37 \( 1 + 226 T + p^{3} T^{2} \)
41 \( 1 + 90 T + p^{3} T^{2} \)
43 \( 1 + 452 T + p^{3} T^{2} \)
47 \( 1 - 432 T + p^{3} T^{2} \)
53 \( 1 - 414 T + p^{3} T^{2} \)
59 \( 1 + 684 T + p^{3} T^{2} \)
61 \( 1 + 422 T + p^{3} T^{2} \)
67 \( 1 + 332 T + p^{3} T^{2} \)
71 \( 1 - 360 T + p^{3} T^{2} \)
73 \( 1 + 26 T + p^{3} T^{2} \)
79 \( 1 + 512 T + p^{3} T^{2} \)
83 \( 1 + 1188 T + p^{3} T^{2} \)
89 \( 1 - 630 T + p^{3} T^{2} \)
97 \( 1 - 1054 T + p^{3} T^{2} \)
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

−8.435345353318372028511561445344, −7.50157172566255003554723165140, −6.63479302052772567471985878571, −5.85477409376154177898259317008, −5.24193842140678374339744308941, −4.21063083045158008794224234532, −3.08238486225049853354590263800, −2.15543935618842266241481588059, −1.67558855992521747268231666010, 0, 1.67558855992521747268231666010, 2.15543935618842266241481588059, 3.08238486225049853354590263800, 4.21063083045158008794224234532, 5.24193842140678374339744308941, 5.85477409376154177898259317008, 6.63479302052772567471985878571, 7.50157172566255003554723165140, 8.435345353318372028511561445344

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