Properties

Label 2-18240-1.1-c1-0-15
Degree $2$
Conductor $18240$
Sign $1$
Analytic cond. $145.647$
Root an. cond. $12.0684$
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 + 5-s + 9-s − 4·11-s − 2·13-s + 15-s + 2·17-s + 19-s + 4·23-s + 25-s + 27-s − 6·29-s + 4·31-s − 4·33-s + 6·37-s − 2·39-s + 10·41-s + 4·43-s + 45-s − 12·47-s − 7·49-s + 2·51-s − 6·53-s − 4·55-s + 57-s + 12·59-s + 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s − 49-s + 0.280·51-s − 0.824·53-s − 0.539·55-s + 0.132·57-s + 1.56·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18240\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(145.647\)
Root analytic conductor: \(12.0684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{18240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 18240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.705277752\)
\(L(\frac12)\) \(\approx\) \(2.705277752\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−15.89464989295771, −15.07816321685752, −14.59442088986342, −14.37033066069411, −13.32316980120882, −13.22489445823221, −12.68394389881669, −12.01402825732025, −11.14554627873736, −10.85763009564661, −9.910435922832978, −9.735616207679410, −9.118133164230526, −8.317987874729830, −7.779942748725993, −7.401982058930073, −6.577293129709557, −5.882188157079488, −5.169727201994963, −4.731931019718075, −3.807086926759255, −2.955141008483067, −2.552641598593329, −1.704472432998404, −0.6616482933578650, 0.6616482933578650, 1.704472432998404, 2.552641598593329, 2.955141008483067, 3.807086926759255, 4.731931019718075, 5.169727201994963, 5.882188157079488, 6.577293129709557, 7.401982058930073, 7.779942748725993, 8.317987874729830, 9.118133164230526, 9.735616207679410, 9.910435922832978, 10.85763009564661, 11.14554627873736, 12.01402825732025, 12.68394389881669, 13.22489445823221, 13.32316980120882, 14.37033066069411, 14.59442088986342, 15.07816321685752, 15.89464989295771

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