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Properties

Label 2-92400-1.1-c1-0-133
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 − 4·13-s + 2·17-s − 21-s − 6·23-s + 27-s + 4·29-s − 2·31-s + 33-s + 4·37-s − 4·39-s − 10·41-s − 4·43-s + 49-s + 2·51-s − 6·53-s − 4·59-s − 2·61-s − 63-s + 8·67-s − 6·69-s + 8·71-s − 77-s + 14·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.485·17-s − 0.218·21-s − 1.25·23-s + 0.192·27-s + 0.742·29-s − 0.359·31-s + 0.174·33-s + 0.657·37-s − 0.640·39-s − 1.56·41-s − 0.609·43-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.520·59-s − 0.256·61-s − 0.125·63-s + 0.977·67-s − 0.722·69-s + 0.949·71-s − 0.113·77-s + 1.57·79-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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.11132857792278, −13.71216021075871, −13.13487274330623, −12.50406470935267, −12.24682456711560, −11.75557695760355, −11.16930111235027, −10.42804347684895, −10.01521335125620, −9.662409358009592, −9.205934831069574, −8.509441741149358, −8.072174035116429, −7.590157194529489, −7.070129612366901, −6.401262194621031, −6.107365537875392, −5.117735746729660, −4.882693565424756, −4.082639051061129, −3.508507270458066, −3.055686612794210, −2.245498518031316, −1.853486296085977, −0.8793568693773170, 0, 0.8793568693773170, 1.853486296085977, 2.245498518031316, 3.055686612794210, 3.508507270458066, 4.082639051061129, 4.882693565424756, 5.117735746729660, 6.107365537875392, 6.401262194621031, 7.070129612366901, 7.590157194529489, 8.072174035116429, 8.509441741149358, 9.205934831069574, 9.662409358009592, 10.01521335125620, 10.42804347684895, 11.16930111235027, 11.75557695760355, 12.24682456711560, 12.50406470935267, 13.13487274330623, 13.71216021075871, 14.11132857792278

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