Properties

Label 2-9240-1.1-c1-0-5
Degree $2$
Conductor $9240$
Sign $1$
Analytic cond. $73.7817$
Root an. cond. $8.58963$
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 + 7-s + 9-s − 11-s + 2·13-s + 15-s − 6·17-s − 4·19-s − 21-s − 8·23-s + 25-s − 27-s − 6·29-s + 4·31-s + 33-s − 35-s + 2·37-s − 2·39-s − 6·41-s + 12·43-s − 45-s + 49-s + 6·51-s + 2·53-s + 55-s + 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.169·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s + 0.274·53-s + 0.134·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(73.7817\)
Root analytic conductor: \(8.58963\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9091010731\)
\(L(\frac12)\) \(\approx\) \(0.9091010731\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−7.70100095091290825022096262556, −7.03471564348879185663033571990, −6.19845162855533512646462735970, −5.86260375849636690024930073817, −4.81484967856810204655145614778, −4.27784002976064669754156595151, −3.70448934051116977633409995430, −2.44920194318402002902520351720, −1.76954475875817266771400278922, −0.46033672039814631545460229320, 0.46033672039814631545460229320, 1.76954475875817266771400278922, 2.44920194318402002902520351720, 3.70448934051116977633409995430, 4.27784002976064669754156595151, 4.81484967856810204655145614778, 5.86260375849636690024930073817, 6.19845162855533512646462735970, 7.03471564348879185663033571990, 7.70100095091290825022096262556

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