Properties

Label 2-58800-1.1-c1-0-50
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

\(L(1)\) \(\approx\) \(2.480550666\)
\(L(\frac12)\) \(\approx\) \(2.480550666\)
\(L(\frac{3}{2})\) 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

−14.20802103165076, −13.88869927339139, −13.22895304939640, −13.06653701202259, −12.31570052293962, −11.90323349255542, −11.14106267139671, −10.79946287700176, −10.23528632567087, −9.675019919298945, −9.123130563725544, −8.656364083073491, −8.139334206248246, −7.623321545447822, −7.021703346247681, −6.562698606197410, −5.769714517133071, −5.349183205204009, −4.492078729969340, −4.146178299000131, −3.358942303759547, −2.740820430036227, −2.204602812233934, −1.434131695142258, −0.5176873653620805, 0.5176873653620805, 1.434131695142258, 2.204602812233934, 2.740820430036227, 3.358942303759547, 4.146178299000131, 4.492078729969340, 5.349183205204009, 5.769714517133071, 6.562698606197410, 7.021703346247681, 7.623321545447822, 8.139334206248246, 8.656364083073491, 9.123130563725544, 9.675019919298945, 10.23528632567087, 10.79946287700176, 11.14106267139671, 11.90323349255542, 12.31570052293962, 13.06653701202259, 13.22895304939640, 13.88869927339139, 14.20802103165076

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