Properties

Label 2-9240-1.1-c1-0-112
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 $1$

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 + 25-s + 27-s + 6·29-s − 33-s + 35-s − 10·37-s − 2·39-s + 2·41-s + 4·43-s + 45-s + 49-s − 6·51-s + 6·53-s − 55-s − 4·57-s − 4·59-s − 2·61-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 + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.174·33-s + 0.169·35-s − 1.64·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.134·55-s − 0.529·57-s − 0.520·59-s − 0.256·61-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: \(1\)
Selberg data: \((2,\ 9240,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−7.34035938791481623686709703956, −6.77004519568473908591243407285, −6.10862618715414707712939278910, −5.17794744804159747297140173809, −4.57177344100419345188825782545, −3.92942687079285507701656543942, −2.78390516829256487687830218675, −2.30580985801589299849220239329, −1.45764992325430760755612977982, 0, 1.45764992325430760755612977982, 2.30580985801589299849220239329, 2.78390516829256487687830218675, 3.92942687079285507701656543942, 4.57177344100419345188825782545, 5.17794744804159747297140173809, 6.10862618715414707712939278910, 6.77004519568473908591243407285, 7.34035938791481623686709703956

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