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Properties

Label 2-92400-1.1-c1-0-13
Degree $2$
Conductor $92400$
Sign $1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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 − 2·13-s − 6·17-s + 8·19-s + 21-s + 4·23-s − 27-s + 2·29-s − 8·31-s + 33-s − 6·37-s + 2·39-s + 6·41-s + 8·43-s + 4·47-s + 49-s + 6·51-s − 10·53-s − 8·57-s − 4·59-s − 14·61-s − 63-s − 4·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 1.45·17-s + 1.83·19-s + 0.218·21-s + 0.834·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.174·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 1.05·57-s − 0.520·59-s − 1.79·61-s − 0.125·63-s − 0.488·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9807363003\)
\(L(\frac12)\) \(\approx\) \(0.9807363003\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.82811882162912, −13.38956905532693, −12.70157379147790, −12.35435950766865, −12.04210308187245, −11.16822880449045, −10.88435129830411, −10.68155885407516, −9.668774418026871, −9.398441834202158, −9.119024578720419, −8.273466672914789, −7.612976231767010, −7.237994433833008, −6.764506794545232, −6.203868669785465, −5.435550199817182, −5.252544545695133, −4.518628292783235, −4.012117198076873, −3.168108409928034, −2.755490247573199, −1.942129755540114, −1.191003853296859, −0.3467843110438065, 0.3467843110438065, 1.191003853296859, 1.942129755540114, 2.755490247573199, 3.168108409928034, 4.012117198076873, 4.518628292783235, 5.252544545695133, 5.435550199817182, 6.203868669785465, 6.764506794545232, 7.237994433833008, 7.612976231767010, 8.273466672914789, 9.119024578720419, 9.398441834202158, 9.668774418026871, 10.68155885407516, 10.88435129830411, 11.16822880449045, 12.04210308187245, 12.35435950766865, 12.70157379147790, 13.38956905532693, 13.82811882162912

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