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Properties

Label 2-92400-1.1-c1-0-186
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 − 2·13-s − 6·17-s + 4·19-s + 21-s + 27-s + 6·29-s + 4·31-s + 33-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s + 49-s − 6·51-s − 6·53-s + 4·57-s + 2·61-s + 63-s − 4·67-s − 2·73-s + 77-s − 8·79-s + 81-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 + 0.917·19-s + 0.218·21-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 0.256·61-s + 0.125·63-s − 0.488·67-s − 0.234·73-s + 0.113·77-s − 0.900·79-s + 1/9·81-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.06825155089978, −13.65166680135449, −13.22869136182023, −12.65100013925510, −12.02415022033563, −11.75361995450035, −11.12010695354490, −10.59886258505221, −10.11601314158445, −9.453368319628831, −9.180630107661431, −8.561194035247734, −8.084615220852419, −7.630962370073023, −6.938156639525405, −6.620793980030408, −5.972326983512323, −5.165259388824792, −4.727672624944578, −4.235471397931209, −3.600485929851459, −2.785769880002965, −2.487717625121072, −1.650998568202022, −1.025776771358898, 0, 1.025776771358898, 1.650998568202022, 2.487717625121072, 2.785769880002965, 3.600485929851459, 4.235471397931209, 4.727672624944578, 5.165259388824792, 5.972326983512323, 6.620793980030408, 6.938156639525405, 7.630962370073023, 8.084615220852419, 8.561194035247734, 9.180630107661431, 9.453368319628831, 10.11601314158445, 10.59886258505221, 11.12010695354490, 11.75361995450035, 12.02415022033563, 12.65100013925510, 13.22869136182023, 13.65166680135449, 14.06825155089978

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