Properties

Label 2-235200-1.1-c1-0-229
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 $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

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: $\chi_{235200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.643456125\)
\(L(\frac12)\) \(\approx\) \(2.643456125\)
\(L(\frac{3}{2})\) 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

−12.81839576266081, −12.37088992545227, −11.90003135029057, −11.50745203800990, −11.18849977005457, −10.77144944881547, −9.985363351148373, −9.572815046300739, −9.359378764571290, −8.780028040590361, −8.163696944126345, −7.729814842006905, −6.899771531221196, −6.827547277719122, −6.269702617977023, −5.803874658150116, −5.113490495678135, −4.671731117737893, −4.222067491894227, −3.581892750864101, −3.145259681767742, −2.324443705014300, −1.671168370753898, −1.087915648828132, −0.5290392800752154, 0.5290392800752154, 1.087915648828132, 1.671168370753898, 2.324443705014300, 3.145259681767742, 3.581892750864101, 4.222067491894227, 4.671731117737893, 5.113490495678135, 5.803874658150116, 6.269702617977023, 6.827547277719122, 6.899771531221196, 7.729814842006905, 8.163696944126345, 8.780028040590361, 9.359378764571290, 9.572815046300739, 9.985363351148373, 10.77144944881547, 11.18849977005457, 11.50745203800990, 11.90003135029057, 12.37088992545227, 12.81839576266081

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