Properties

Label 2-369600-1.1-c1-0-130
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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 + 7-s + 9-s − 11-s + 7·13-s + 3·17-s − 19-s − 21-s + 4·23-s − 27-s − 8·29-s − 6·31-s + 33-s + 7·37-s − 7·39-s − 9·41-s − 6·43-s + 2·47-s + 49-s − 3·51-s + 3·53-s + 57-s + 14·59-s − 3·61-s + 63-s − 5·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.94·13-s + 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s − 1.48·29-s − 1.07·31-s + 0.174·33-s + 1.15·37-s − 1.12·39-s − 1.40·41-s − 0.914·43-s + 0.291·47-s + 1/7·49-s − 0.420·51-s + 0.412·53-s + 0.132·57-s + 1.82·59-s − 0.384·61-s + 0.125·63-s − 0.610·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.376999262\)
\(L(\frac12)\) \(\approx\) \(2.376999262\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.61959443003241, −11.89870318994087, −11.46772690887382, −11.18651245725674, −10.83591542370758, −10.33181500131491, −9.894008754591832, −9.267508936145016, −8.877893643053314, −8.340552395340641, −8.020669435625334, −7.371895921238356, −6.977085798648079, −6.442906247028624, −5.900720694280754, −5.501049618136210, −5.193413506753086, −4.509956011737354, −3.836579046893385, −3.607496569150947, −3.008544782883084, −2.149967783750685, −1.600319135639052, −1.116778874496764, −0.4512023749204111, 0.4512023749204111, 1.116778874496764, 1.600319135639052, 2.149967783750685, 3.008544782883084, 3.607496569150947, 3.836579046893385, 4.509956011737354, 5.193413506753086, 5.501049618136210, 5.900720694280754, 6.442906247028624, 6.977085798648079, 7.371895921238356, 8.020669435625334, 8.340552395340641, 8.877893643053314, 9.267508936145016, 9.894008754591832, 10.33181500131491, 10.83591542370758, 11.18651245725674, 11.46772690887382, 11.89870318994087, 12.61959443003241

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