Properties

Degree $2$
Conductor $82800$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s + 2·13-s − 6·17-s − 4·19-s + 23-s + 2·29-s + 2·37-s − 10·41-s − 4·43-s − 7·49-s + 6·53-s − 4·59-s − 10·61-s − 12·67-s − 8·71-s − 10·73-s + 8·79-s + 4·83-s − 18·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.208·23-s + 0.371·29-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s + 0.824·53-s − 0.520·59-s − 1.28·61-s − 1.46·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s + 0.439·83-s − 1.90·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82800\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{82800} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 82800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.571879264\)
\(L(\frac12)\) \(\approx\) \(1.571879264\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.82324917889207, −13.43244448321799, −13.14374608675127, −12.42292467301843, −11.88173500610804, −11.58186046667871, −10.87821831073782, −10.64777883551879, −9.937271472938617, −9.361387342229931, −8.799217848267564, −8.607074595978768, −7.975042790067582, −7.172790008847788, −6.680950678462847, −6.370240169260920, −5.829984232632385, −5.003921754574432, −4.350917671088369, −4.142557085999881, −3.287210112000598, −2.765816156836010, −1.756836036669573, −1.531046776178817, −0.3953555816073852, 0.3953555816073852, 1.531046776178817, 1.756836036669573, 2.765816156836010, 3.287210112000598, 4.142557085999881, 4.350917671088369, 5.003921754574432, 5.829984232632385, 6.370240169260920, 6.680950678462847, 7.172790008847788, 7.975042790067582, 8.607074595978768, 8.799217848267564, 9.361387342229931, 9.937271472938617, 10.64777883551879, 10.87821831073782, 11.58186046667871, 11.88173500610804, 12.42292467301843, 13.14374608675127, 13.43244448321799, 13.82324917889207

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