Properties

Label 2-25200-1.1-c1-0-68
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 + 6·11-s + 13-s + 3·17-s + 4·19-s + 3·23-s − 3·29-s − 5·31-s + 10·37-s − 9·41-s − 43-s + 49-s + 9·53-s + 9·59-s + 11·61-s − 4·67-s − 12·71-s + 10·73-s + 6·77-s + 10·79-s − 9·83-s + 6·89-s + 91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.80·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s + 0.625·23-s − 0.557·29-s − 0.898·31-s + 1.64·37-s − 1.40·41-s − 0.152·43-s + 1/7·49-s + 1.23·53-s + 1.17·59-s + 1.40·61-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.683·77-s + 1.12·79-s − 0.987·83-s + 0.635·89-s + 0.104·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(3.405094989\)
\(L(\frac12)\) \(\approx\) \(3.405094989\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−15.16749242857127, −14.63435301563257, −14.55199977580649, −13.75405174238478, −13.29154834956472, −12.67011057164665, −11.90684998147792, −11.64150403583089, −11.21953349187978, −10.45610368516423, −9.752863841814162, −9.370786561835511, −8.774030728287253, −8.244993876985157, −7.446281039302665, −7.020733064821601, −6.390382701478395, −5.666919860268074, −5.195202222115868, −4.324678216224819, −3.733114121738830, −3.247466186051301, −2.199106002653414, −1.368088057924999, −0.8297656693209773, 0.8297656693209773, 1.368088057924999, 2.199106002653414, 3.247466186051301, 3.733114121738830, 4.324678216224819, 5.195202222115868, 5.666919860268074, 6.390382701478395, 7.020733064821601, 7.446281039302665, 8.244993876985157, 8.774030728287253, 9.370786561835511, 9.752863841814162, 10.45610368516423, 11.21953349187978, 11.64150403583089, 11.90684998147792, 12.67011057164665, 13.29154834956472, 13.75405174238478, 14.55199977580649, 14.63435301563257, 15.16749242857127

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