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Properties

Label 2-92400-1.1-c1-0-14
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 − 13-s − 6·17-s − 5·19-s + 21-s + 3·23-s − 27-s + 3·29-s − 5·31-s − 33-s + 8·37-s + 39-s + 43-s + 49-s + 6·51-s + 5·57-s + 2·61-s − 63-s + 4·67-s − 3·69-s + 3·71-s + 8·73-s − 77-s + 10·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 1.45·17-s − 1.14·19-s + 0.218·21-s + 0.625·23-s − 0.192·27-s + 0.557·29-s − 0.898·31-s − 0.174·33-s + 1.31·37-s + 0.160·39-s + 0.152·43-s + 1/7·49-s + 0.840·51-s + 0.662·57-s + 0.256·61-s − 0.125·63-s + 0.488·67-s − 0.361·69-s + 0.356·71-s + 0.936·73-s − 0.113·77-s + 1.12·79-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

\(L(1)\) \(\approx\) \(1.004107460\)
\(L(\frac12)\) \(\approx\) \(1.004107460\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−13.86577121875217, −13.08321602267660, −12.85623243122103, −12.56092733799893, −11.74357299215902, −11.37441515784553, −10.90418891512976, −10.50762977254976, −9.897897881777187, −9.299785445246992, −8.973392277645230, −8.351035535804544, −7.791584919387014, −7.000140249659575, −6.751147838140747, −6.235981939391893, −5.714740324675456, −4.994067528989941, −4.484028600576568, −4.048521274379735, −3.348972715970215, −2.474937523852297, −2.107449181266854, −1.154911012413924, −0.3539203952538713, 0.3539203952538713, 1.154911012413924, 2.107449181266854, 2.474937523852297, 3.348972715970215, 4.048521274379735, 4.484028600576568, 4.994067528989941, 5.714740324675456, 6.235981939391893, 6.751147838140747, 7.000140249659575, 7.791584919387014, 8.351035535804544, 8.973392277645230, 9.299785445246992, 9.897897881777187, 10.50762977254976, 10.90418891512976, 11.37441515784553, 11.74357299215902, 12.56092733799893, 12.85623243122103, 13.08321602267660, 13.86577121875217

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