Properties

Label 2-58800-1.1-c1-0-103
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 6·11-s − 13-s + 3·17-s − 4·19-s − 3·23-s − 27-s + 3·29-s + 5·31-s + 6·33-s + 10·37-s + 39-s − 9·41-s − 43-s − 3·51-s − 9·53-s + 4·57-s + 9·59-s − 11·61-s − 4·67-s + 3·69-s + 12·71-s − 10·73-s + 10·79-s + 81-s − 9·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s − 0.625·23-s − 0.192·27-s + 0.557·29-s + 0.898·31-s + 1.04·33-s + 1.64·37-s + 0.160·39-s − 1.40·41-s − 0.152·43-s − 0.420·51-s − 1.23·53-s + 0.529·57-s + 1.17·59-s − 1.40·61-s − 0.488·67-s + 0.361·69-s + 1.42·71-s − 1.17·73-s + 1.12·79-s + 1/9·81-s − 0.987·83-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: \(1\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ -1)\)

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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
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

−14.73002986022611, −13.96614208805759, −13.52564046452589, −12.98455798215748, −12.58105180403185, −12.12726465039134, −11.49959582172637, −11.03908445910568, −10.40988829594765, −10.05194826041469, −9.782560394805546, −8.784761774256016, −8.320122262185663, −7.650468643827094, −7.551878106521224, −6.472145799766786, −6.223492727452183, −5.551840393215168, −4.872167163134722, −4.679135654625971, −3.787783994139921, −3.001051566345191, −2.477158553851844, −1.755004674222598, −0.7469061219077777, 0, 0.7469061219077777, 1.755004674222598, 2.477158553851844, 3.001051566345191, 3.787783994139921, 4.679135654625971, 4.872167163134722, 5.551840393215168, 6.223492727452183, 6.472145799766786, 7.551878106521224, 7.650468643827094, 8.320122262185663, 8.784761774256016, 9.782560394805546, 10.05194826041469, 10.40988829594765, 11.03908445910568, 11.49959582172637, 12.12726465039134, 12.58105180403185, 12.98455798215748, 13.52564046452589, 13.96614208805759, 14.73002986022611

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