Properties

Label 2-58800-1.1-c1-0-17
Degree $2$
Conductor $58800$
Sign $1$
Analytic cond. $469.520$
Root an. cond. $21.6684$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 2·11-s − 13-s + 3·17-s − 23-s − 27-s − 5·29-s + 7·31-s + 2·33-s + 2·37-s + 39-s − 7·41-s − 11·43-s − 8·47-s − 3·51-s + 53-s − 5·59-s + 3·61-s − 12·67-s + 69-s − 12·71-s − 6·73-s − 10·79-s + 81-s + 11·83-s + 5·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.208·23-s − 0.192·27-s − 0.928·29-s + 1.25·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s − 1.09·41-s − 1.67·43-s − 1.16·47-s − 0.420·51-s + 0.137·53-s − 0.650·59-s + 0.384·61-s − 1.46·67-s + 0.120·69-s − 1.42·71-s − 0.702·73-s − 1.12·79-s + 1/9·81-s + 1.20·83-s + 0.536·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(469.520\)
Root analytic conductor: \(21.6684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8874687858\)
\(L(\frac12)\) \(\approx\) \(0.8874687858\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
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

−14.42624968651241, −13.67961754081023, −13.23245951810758, −12.95398389261853, −12.11238116612490, −11.82221153872033, −11.46113212671618, −10.64273378802419, −10.26259978392624, −9.894625963728385, −9.281661099314938, −8.581977844599842, −8.008036356938591, −7.606271849888018, −6.940818012107432, −6.407949581673077, −5.823205516347204, −5.272699055129798, −4.772874754009128, −4.209002325544695, −3.310798903936702, −2.930735015600565, −1.936313612052946, −1.369536720968158, −0.3373018454497730, 0.3373018454497730, 1.369536720968158, 1.936313612052946, 2.930735015600565, 3.310798903936702, 4.209002325544695, 4.772874754009128, 5.272699055129798, 5.823205516347204, 6.407949581673077, 6.940818012107432, 7.606271849888018, 8.008036356938591, 8.581977844599842, 9.281661099314938, 9.894625963728385, 10.26259978392624, 10.64273378802419, 11.46113212671618, 11.82221153872033, 12.11238116612490, 12.95398389261853, 13.23245951810758, 13.67961754081023, 14.42624968651241

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