↑

Properties

Label 2-92400-1.1-c1-0-119
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s + 11-s + 6·13-s + 2·17-s − 4·19-s + 21-s + 8·23-s + 27-s − 2·29-s + 33-s + 10·37-s + 6·39-s + 2·41-s + 8·43-s + 49-s + 2·51-s + 10·53-s − 4·57-s − 4·59-s + 2·61-s + 63-s + 12·67-s + 8·69-s − 4·71-s − 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 0.371·29-s + 0.174·33-s + 1.64·37-s + 0.960·39-s + 0.312·41-s + 1.21·43-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.125·63-s + 1.46·67-s + 0.963·69-s − 0.474·71-s − 0.702·73-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.245312721\)
\(L(\frac12)\) \(\approx\) \(5.245312721\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−13.77788838587282, −13.35444946992905, −12.96328721032771, −12.52730618901565, −11.82533011180075, −11.26572032531040, −10.81435305413699, −10.62212636623754, −9.688747574253677, −9.315972425528891, −8.712844518134205, −8.497732105421528, −7.821666963432197, −7.367401033270042, −6.734552844115751, −6.144744561091882, −5.756114353403596, −4.968422677939860, −4.388071017168300, −3.809478825139900, −3.396058726014361, −2.604329487558144, −2.062904630100040, −1.130423059577806, −0.8433415912489656, 0.8433415912489656, 1.130423059577806, 2.062904630100040, 2.604329487558144, 3.396058726014361, 3.809478825139900, 4.388071017168300, 4.968422677939860, 5.756114353403596, 6.144744561091882, 6.734552844115751, 7.367401033270042, 7.821666963432197, 8.497732105421528, 8.712844518134205, 9.315972425528891, 9.688747574253677, 10.62212636623754, 10.81435305413699, 11.26572032531040, 11.82533011180075, 12.52730618901565, 12.96328721032771, 13.35444946992905, 13.77788838587282

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