Properties

Label 2-92400-1.1-c1-0-110
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 + 4·13-s − 4·17-s − 4·19-s + 21-s − 6·23-s − 27-s + 10·29-s + 33-s + 8·37-s − 4·39-s + 6·41-s + 8·47-s + 49-s + 4·51-s − 10·53-s + 4·57-s + 8·59-s − 10·61-s − 63-s − 14·67-s + 6·69-s + 12·71-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.970·17-s − 0.917·19-s + 0.218·21-s − 1.25·23-s − 0.192·27-s + 1.85·29-s + 0.174·33-s + 1.31·37-s − 0.640·39-s + 0.937·41-s + 1.16·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s + 0.529·57-s + 1.04·59-s − 1.28·61-s − 0.125·63-s − 1.71·67-s + 0.722·69-s + 1.42·71-s + 1.17·73-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−14.01855421277902, −13.56897709483942, −13.02780737916587, −12.60586997434566, −12.22154889772216, −11.49707324241192, −11.16428268455153, −10.63704691302628, −10.23368412120914, −9.745763797085892, −8.950380828986767, −8.730586659402417, −7.951400047234529, −7.673878526483283, −6.684888504280610, −6.432250244083661, −6.034305673695014, −5.487088605468291, −4.579722413714464, −4.308263264270169, −3.769786153666419, −2.844806748082646, −2.394451613837050, −1.551206011130931, −0.7934500827082348, 0, 0.7934500827082348, 1.551206011130931, 2.394451613837050, 2.844806748082646, 3.769786153666419, 4.308263264270169, 4.579722413714464, 5.487088605468291, 6.034305673695014, 6.432250244083661, 6.684888504280610, 7.673878526483283, 7.951400047234529, 8.730586659402417, 8.950380828986767, 9.745763797085892, 10.23368412120914, 10.63704691302628, 11.16428268455153, 11.49707324241192, 12.22154889772216, 12.60586997434566, 13.02780737916587, 13.56897709483942, 14.01855421277902

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