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Properties

Label 2-92400-1.1-c1-0-190
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 11-s − 13-s + 5·17-s + 5·19-s − 21-s + 8·23-s + 27-s − 6·29-s − 8·31-s − 33-s + 11·37-s − 39-s − 5·41-s + 4·43-s + 4·47-s + 49-s + 5·51-s + 3·53-s + 5·57-s − 14·59-s − 5·61-s − 63-s + 3·67-s + 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 1.21·17-s + 1.14·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.174·33-s + 1.80·37-s − 0.160·39-s − 0.780·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.700·51-s + 0.412·53-s + 0.662·57-s − 1.82·59-s − 0.640·61-s − 0.125·63-s + 0.366·67-s + 0.963·69-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: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.14534758603401, −13.57874720489109, −13.01986367343340, −12.78973819600895, −12.21975175229489, −11.63149503059246, −11.07306542661712, −10.66634062190583, −9.958796789669853, −9.546156774728265, −9.189853149442091, −8.715016920918117, −7.893704883772824, −7.419042781975679, −7.341242608700894, −6.519909886356769, −5.708694014466464, −5.458046787929301, −4.802306964664563, −4.069771343757917, −3.465940066365609, −2.964819188419492, −2.548806762312773, −1.527200199340632, −1.052372374163079, 0, 1.052372374163079, 1.527200199340632, 2.548806762312773, 2.964819188419492, 3.465940066365609, 4.069771343757917, 4.802306964664563, 5.458046787929301, 5.708694014466464, 6.519909886356769, 7.341242608700894, 7.419042781975679, 7.893704883772824, 8.715016920918117, 9.189853149442091, 9.546156774728265, 9.958796789669853, 10.66634062190583, 11.07306542661712, 11.63149503059246, 12.21975175229489, 12.78973819600895, 13.01986367343340, 13.57874720489109, 14.14534758603401

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