Properties

Label 2-25200-1.1-c1-0-20
Degree $2$
Conductor $25200$
Sign $1$
Analytic cond. $201.223$
Root an. cond. $14.1853$
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  + 7-s − 3·11-s − 5·13-s + 3·17-s − 2·19-s + 6·23-s − 3·29-s + 4·31-s − 2·37-s + 12·41-s − 10·43-s − 9·47-s + 49-s + 12·53-s + 8·61-s − 4·67-s − 2·73-s − 3·77-s + 79-s − 12·83-s + 12·89-s − 5·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.904·11-s − 1.38·13-s + 0.727·17-s − 0.458·19-s + 1.25·23-s − 0.557·29-s + 0.718·31-s − 0.328·37-s + 1.87·41-s − 1.52·43-s − 1.31·47-s + 1/7·49-s + 1.64·53-s + 1.02·61-s − 0.488·67-s − 0.234·73-s − 0.341·77-s + 0.112·79-s − 1.31·83-s + 1.27·89-s − 0.524·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(201.223\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{25200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.633418113\)
\(L(\frac12)\) \(\approx\) \(1.633418113\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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

−15.16948026538744, −14.84546660163484, −14.46418275446968, −13.75656305585832, −13.01772771928668, −12.85120427098183, −12.10313267096716, −11.58615329015504, −11.07419200230179, −10.22392507156937, −10.14205733920058, −9.328253216627123, −8.745892417642117, −8.041709351011833, −7.609671816180865, −7.066596918043708, −6.422026442864954, −5.478989364024287, −5.158201832164046, −4.580727044941540, −3.773052595184903, −2.853862179976577, −2.464949668978444, −1.521017786445178, −0.5040556459025151, 0.5040556459025151, 1.521017786445178, 2.464949668978444, 2.853862179976577, 3.773052595184903, 4.580727044941540, 5.158201832164046, 5.478989364024287, 6.422026442864954, 7.066596918043708, 7.609671816180865, 8.041709351011833, 8.745892417642117, 9.328253216627123, 10.14205733920058, 10.22392507156937, 11.07419200230179, 11.58615329015504, 12.10313267096716, 12.85120427098183, 13.01772771928668, 13.75656305585832, 14.46418275446968, 14.84546660163484, 15.16948026538744

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