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Properties

Label 2-92400-1.1-c1-0-130
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 − 4·17-s − 3·19-s − 21-s + 6·23-s + 27-s + 7·29-s − 4·31-s − 33-s − 37-s − 39-s − 4·41-s − 2·43-s + 7·47-s + 49-s − 4·51-s − 10·53-s − 3·57-s + 9·59-s − 2·61-s − 63-s − 9·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.970·17-s − 0.688·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s + 1.29·29-s − 0.718·31-s − 0.174·33-s − 0.164·37-s − 0.160·39-s − 0.624·41-s − 0.304·43-s + 1.02·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s − 0.397·57-s + 1.17·59-s − 0.256·61-s − 0.125·63-s − 1.09·67-s + 0.722·69-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 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−14.01370278754570, −13.59877559734233, −13.06221978538405, −12.76281849845383, −12.25422864339068, −11.62315049101158, −11.04526812070805, −10.56878519767891, −10.18102820633211, −9.509424733089351, −9.061468611394850, −8.622569680028976, −8.181839201625462, −7.510568216769700, −6.896264883938588, −6.679605556701650, −5.949908433340466, −5.269508161328750, −4.649119876681783, −4.278720704658409, −3.441064188465310, −2.983123154406360, −2.369167171448963, −1.793932624710756, −0.8756922955429957, 0, 0.8756922955429957, 1.793932624710756, 2.369167171448963, 2.983123154406360, 3.441064188465310, 4.278720704658409, 4.649119876681783, 5.269508161328750, 5.949908433340466, 6.679605556701650, 6.896264883938588, 7.510568216769700, 8.181839201625462, 8.622569680028976, 9.061468611394850, 9.509424733089351, 10.18102820633211, 10.56878519767891, 11.04526812070805, 11.62315049101158, 12.25422864339068, 12.76281849845383, 13.06221978538405, 13.59877559734233, 14.01370278754570

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