Properties

Degree $2$
Conductor $25200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 4·11-s + 6·13-s + 2·17-s − 6·29-s − 8·31-s + 10·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 2·53-s − 8·59-s − 14·61-s − 12·67-s − 16·71-s − 2·73-s − 4·77-s + 8·79-s − 8·83-s − 10·89-s − 6·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 1.04·59-s − 1.79·61-s − 1.46·67-s − 1.89·71-s − 0.234·73-s − 0.455·77-s + 0.900·79-s − 0.878·83-s − 1.05·89-s − 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{25200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.67225083859428, −14.94644561349370, −14.67720168082641, −13.97453481354031, −13.49527811742600, −12.93732035748255, −12.51614036892032, −11.72259650091484, −11.30720593768393, −10.87082540907919, −10.18134088651184, −9.453342363461301, −9.046144138277398, −8.666270407425560, −7.687415713700300, −7.426673015063221, −6.404926253783660, −6.147463424058473, −5.633641894516340, −4.641112214827860, −3.958304595354019, −3.520408916186592, −2.849232259202210, −1.603751430783516, −1.289536795897307, 0, 1.289536795897307, 1.603751430783516, 2.849232259202210, 3.520408916186592, 3.958304595354019, 4.641112214827860, 5.633641894516340, 6.147463424058473, 6.404926253783660, 7.426673015063221, 7.687415713700300, 8.666270407425560, 9.046144138277398, 9.453342363461301, 10.18134088651184, 10.87082540907919, 11.30720593768393, 11.72259650091484, 12.51614036892032, 12.93732035748255, 13.49527811742600, 13.97453481354031, 14.67720168082641, 14.94644561349370, 15.67225083859428

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