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Properties

Label 2-235200-1.1-c1-0-536
Degree $2$
Conductor $235200$
Sign $-1$
Analytic cond. $1878.08$
Root an. cond. $43.3368$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 2·11-s + 2·13-s − 8·17-s + 2·19-s + 27-s + 6·29-s + 6·31-s − 2·33-s − 8·37-s + 2·39-s − 6·41-s + 8·43-s + 4·47-s − 8·51-s + 2·53-s + 2·57-s + 8·59-s + 10·61-s − 12·67-s + 14·71-s + 10·73-s − 4·79-s + 81-s − 16·83-s + 6·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 1.94·17-s + 0.458·19-s + 0.192·27-s + 1.11·29-s + 1.07·31-s − 0.348·33-s − 1.31·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.583·47-s − 1.12·51-s + 0.274·53-s + 0.264·57-s + 1.04·59-s + 1.28·61-s − 1.46·67-s + 1.66·71-s + 1.17·73-s − 0.450·79-s + 1/9·81-s − 1.75·83-s + 0.643·87-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: $-1$
Analytic conductor: \(1878.08\)
Root analytic conductor: \(43.3368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.20925090150390, −12.80368594278279, −12.20140517054961, −11.83003715643595, −11.16411044756504, −10.84014984635579, −10.36916197998752, −9.845855909313542, −9.431166562931236, −8.733729899043484, −8.409023049656246, −8.287908279668006, −7.371003351224149, −7.001828523078304, −6.580357756788515, −6.059145718495424, −5.294069843296825, −4.987093884497509, −4.161499741390508, −4.038513894997636, −3.188523298407929, −2.615242646089700, −2.307367888234722, −1.524143991654209, −0.8408295523426264, 0, 0.8408295523426264, 1.524143991654209, 2.307367888234722, 2.615242646089700, 3.188523298407929, 4.038513894997636, 4.161499741390508, 4.987093884497509, 5.294069843296825, 6.059145718495424, 6.580357756788515, 7.001828523078304, 7.371003351224149, 8.287908279668006, 8.409023049656246, 8.733729899043484, 9.431166562931236, 9.845855909313542, 10.36916197998752, 10.84014984635579, 11.16411044756504, 11.83003715643595, 12.20140517054961, 12.80368594278279, 13.20925090150390

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