Properties

Degree $2$
Conductor $235200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 5·11-s − 2·13-s − 6·17-s + 2·19-s − 5·23-s − 27-s + 5·29-s − 4·31-s + 5·33-s − 37-s + 2·39-s + 12·41-s + 5·43-s + 2·47-s + 6·51-s − 14·53-s − 2·57-s − 2·59-s − 5·67-s + 5·69-s + 9·71-s − 10·73-s − 11·79-s + 81-s − 16·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s − 1.04·23-s − 0.192·27-s + 0.928·29-s − 0.718·31-s + 0.870·33-s − 0.164·37-s + 0.320·39-s + 1.87·41-s + 0.762·43-s + 0.291·47-s + 0.840·51-s − 1.92·53-s − 0.264·57-s − 0.260·59-s − 0.610·67-s + 0.601·69-s + 1.06·71-s − 1.17·73-s − 1.23·79-s + 1/9·81-s − 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{235200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.18834469588195, −12.90859608853839, −12.45705594569902, −12.07561156265769, −11.35625511653567, −11.11848505532465, −10.56199102096045, −10.30002193461111, −9.634305545756530, −9.314802454357905, −8.651703928407636, −8.151452908436451, −7.582929872017391, −7.343679826696522, −6.724618327240455, −5.963369601275604, −5.913667164672673, −5.111279912084703, −4.708435976501780, −4.309765624188238, −3.655264648024411, −2.714524099142210, −2.570369839167606, −1.846166790442630, −1.066865088420515, 0, 0, 1.066865088420515, 1.846166790442630, 2.570369839167606, 2.714524099142210, 3.655264648024411, 4.309765624188238, 4.708435976501780, 5.111279912084703, 5.913667164672673, 5.963369601275604, 6.724618327240455, 7.343679826696522, 7.582929872017391, 8.151452908436451, 8.651703928407636, 9.314802454357905, 9.634305545756530, 10.30002193461111, 10.56199102096045, 11.11848505532465, 11.35625511653567, 12.07561156265769, 12.45705594569902, 12.90859608853839, 13.18834469588195

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