Properties

Degree $2$
Conductor $185020$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s − 4·7-s + 9-s + 11-s − 4·13-s + 2·15-s + 4·19-s − 8·21-s − 6·23-s + 25-s − 4·27-s − 8·31-s + 2·33-s − 4·35-s − 2·37-s − 8·39-s − 6·41-s − 8·43-s + 45-s − 6·47-s + 9·49-s − 6·53-s + 55-s + 8·57-s − 12·59-s − 2·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.917·19-s − 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.43·31-s + 0.348·33-s − 0.676·35-s − 0.328·37-s − 1.28·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s − 0.875·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s + 1.05·57-s − 1.56·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185020\)    =    \(2^{2} \cdot 5 \cdot 11 \cdot 29^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{185020} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 185020,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 - T \)
11 \( 1 - T \)
29 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.65126961054186, −13.25551203621650, −12.71884808007413, −12.34036514653124, −11.84619494461900, −11.35229361857936, −10.53188696770539, −10.04999434511394, −9.760917406588801, −9.289928505220797, −9.110492625687055, −8.376640195493363, −7.912329460787074, −7.304537700846501, −7.024401586795733, −6.291669518597373, −5.976780456946233, −5.287710543315282, −4.772333653089337, −3.945741234759386, −3.460214277871853, −3.085313795014285, −2.639732647376274, −1.885027437336481, −1.493244884040780, 0, 0, 1.493244884040780, 1.885027437336481, 2.639732647376274, 3.085313795014285, 3.460214277871853, 3.945741234759386, 4.772333653089337, 5.287710543315282, 5.976780456946233, 6.291669518597373, 7.024401586795733, 7.304537700846501, 7.912329460787074, 8.376640195493363, 9.110492625687055, 9.289928505220797, 9.760917406588801, 10.04999434511394, 10.53188696770539, 11.35229361857936, 11.84619494461900, 12.34036514653124, 12.71884808007413, 13.25551203621650, 13.65126961054186

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