Properties

Label 2-9360-1.1-c1-0-35
Degree $2$
Conductor $9360$
Sign $1$
Analytic cond. $74.7399$
Root an. cond. $8.64522$
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  + 5-s − 13-s + 6·17-s − 4·23-s + 25-s + 10·29-s − 6·37-s − 2·41-s + 4·43-s − 7·49-s + 6·53-s + 6·61-s − 65-s − 4·67-s + 16·71-s − 2·73-s + 4·83-s + 6·85-s + 6·89-s + 14·97-s − 6·101-s − 12·103-s − 4·107-s − 14·109-s − 10·113-s − 4·115-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.277·13-s + 1.45·17-s − 0.834·23-s + 1/5·25-s + 1.85·29-s − 0.986·37-s − 0.312·41-s + 0.609·43-s − 49-s + 0.824·53-s + 0.768·61-s − 0.124·65-s − 0.488·67-s + 1.89·71-s − 0.234·73-s + 0.439·83-s + 0.650·85-s + 0.635·89-s + 1.42·97-s − 0.597·101-s − 1.18·103-s − 0.386·107-s − 1.34·109-s − 0.940·113-s − 0.373·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9360\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(74.7399\)
Root analytic conductor: \(8.64522\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.351259045\)
\(L(\frac12)\) \(\approx\) \(2.351259045\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−7.86024313710614696532097430664, −6.90823153056793002056497471386, −6.41051413056067833990175014344, −5.55860505151888914144681124469, −5.10592887722971602042874207321, −4.20725491653826331663773790383, −3.38062750090213723463354695459, −2.63741599748730669205673807869, −1.71741878316152659310548007340, −0.75375237820186639289019479057, 0.75375237820186639289019479057, 1.71741878316152659310548007340, 2.63741599748730669205673807869, 3.38062750090213723463354695459, 4.20725491653826331663773790383, 5.10592887722971602042874207321, 5.55860505151888914144681124469, 6.41051413056067833990175014344, 6.90823153056793002056497471386, 7.86024313710614696532097430664

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