Properties

Label 2-369600-1.1-c1-0-110
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 + 3·13-s − 17-s − 5·19-s + 21-s − 4·23-s − 27-s + 10·31-s − 33-s + 5·37-s − 3·39-s − 3·41-s − 10·43-s + 2·47-s + 49-s + 51-s + 13·53-s + 5·57-s − 6·59-s − 61-s − 63-s − 67-s + 4·69-s + 7·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s − 0.242·17-s − 1.14·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s + 1.79·31-s − 0.174·33-s + 0.821·37-s − 0.480·39-s − 0.468·41-s − 1.52·43-s + 0.291·47-s + 1/7·49-s + 0.140·51-s + 1.78·53-s + 0.662·57-s − 0.781·59-s − 0.128·61-s − 0.125·63-s − 0.122·67-s + 0.481·69-s + 0.830·71-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

\(L(1)\) \(\approx\) \(1.741746374\)
\(L(\frac12)\) \(\approx\) \(1.741746374\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−12.32819543881247, −12.07712420130287, −11.61024583358424, −11.22117508818726, −10.62886796528719, −10.26944957142833, −9.963287284430616, −9.345924593742208, −8.817516281666365, −8.411201804180931, −8.010261211844804, −7.420443377087953, −6.730484164214887, −6.426417017902574, −6.147962391536436, −5.630680432042815, −4.918399163723590, −4.522894865076789, −3.963069197010515, −3.593581326176381, −2.867484361331718, −2.256608502762887, −1.704425534860027, −0.9737394619070005, −0.4099768395616594, 0.4099768395616594, 0.9737394619070005, 1.704425534860027, 2.256608502762887, 2.867484361331718, 3.593581326176381, 3.963069197010515, 4.522894865076789, 4.918399163723590, 5.630680432042815, 6.147962391536436, 6.426417017902574, 6.730484164214887, 7.420443377087953, 8.010261211844804, 8.411201804180931, 8.817516281666365, 9.345924593742208, 9.963287284430616, 10.26944957142833, 10.62886796528719, 11.22117508818726, 11.61024583358424, 12.07712420130287, 12.32819543881247

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