Properties

Label 2-92400-1.1-c1-0-142
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s + 11-s − 13-s + 17-s − 7·19-s − 21-s + 8·23-s − 27-s − 6·29-s − 33-s + 9·37-s + 39-s + 41-s + 8·43-s + 49-s − 51-s + 53-s + 7·57-s + 14·59-s + 5·61-s + 63-s − 5·67-s − 8·69-s − 9·71-s − 7·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.242·17-s − 1.60·19-s − 0.218·21-s + 1.66·23-s − 0.192·27-s − 1.11·29-s − 0.174·33-s + 1.47·37-s + 0.160·39-s + 0.156·41-s + 1.21·43-s + 1/7·49-s − 0.140·51-s + 0.137·53-s + 0.927·57-s + 1.82·59-s + 0.640·61-s + 0.125·63-s − 0.610·67-s − 0.963·69-s − 1.06·71-s − 0.819·73-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: $\chi_{92400} (1, \cdot )$
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 \( 1 - T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.27718352311269, −13.33999054588890, −13.02084618338895, −12.73855416583940, −12.10520837660294, −11.45298414454566, −11.21541717193725, −10.74956346197842, −10.19180710551137, −9.665363063210624, −9.039191165809081, −8.678202304226500, −8.059527013013715, −7.333101216721690, −7.092416425332388, −6.426051501124834, −5.803766774268132, −5.476121124599132, −4.692772368113652, −4.282176693621768, −3.793850023717253, −2.829681134008645, −2.357802055778870, −1.511527217266322, −0.8949810674446571, 0, 0.8949810674446571, 1.511527217266322, 2.357802055778870, 2.829681134008645, 3.793850023717253, 4.282176693621768, 4.692772368113652, 5.476121124599132, 5.803766774268132, 6.426051501124834, 7.092416425332388, 7.333101216721690, 8.059527013013715, 8.678202304226500, 9.039191165809081, 9.665363063210624, 10.19180710551137, 10.74956346197842, 11.21541717193725, 11.45298414454566, 12.10520837660294, 12.73855416583940, 13.02084618338895, 13.33999054588890, 14.27718352311269

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