Properties

Label 2-46800-1.1-c1-0-88
Degree $2$
Conductor $46800$
Sign $-1$
Analytic cond. $373.699$
Root an. cond. $19.3313$
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  − 7-s − 11-s + 13-s − 5·17-s − 3·23-s + 2·29-s + 6·31-s − 37-s + 7·41-s + 2·43-s + 6·47-s − 6·49-s − 5·53-s − 4·59-s − 7·61-s + 12·67-s − 71-s − 10·73-s + 77-s − 11·79-s + 12·83-s + 13·89-s − 91-s + 97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.301·11-s + 0.277·13-s − 1.21·17-s − 0.625·23-s + 0.371·29-s + 1.07·31-s − 0.164·37-s + 1.09·41-s + 0.304·43-s + 0.875·47-s − 6/7·49-s − 0.686·53-s − 0.520·59-s − 0.896·61-s + 1.46·67-s − 0.118·71-s − 1.17·73-s + 0.113·77-s − 1.23·79-s + 1.31·83-s + 1.37·89-s − 0.104·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46800\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(373.699\)
Root analytic conductor: \(19.3313\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{46800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
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

−14.83768763959920, −14.28218060589878, −13.80513481378831, −13.29522545459427, −12.88124400278621, −12.29271610694273, −11.79843819862189, −11.19259025916253, −10.69560944642103, −10.25126860485952, −9.578241429281230, −9.136850936921837, −8.554654944815891, −8.001595455633277, −7.467398667541522, −6.767327096925024, −6.241878457658772, −5.878203166451908, −4.972873229647714, −4.480169449370745, −3.910699271557246, −3.111374213054645, −2.531993408620175, −1.853816018067376, −0.8907578256140091, 0, 0.8907578256140091, 1.853816018067376, 2.531993408620175, 3.111374213054645, 3.910699271557246, 4.480169449370745, 4.972873229647714, 5.878203166451908, 6.241878457658772, 6.767327096925024, 7.467398667541522, 8.001595455633277, 8.554654944815891, 9.136850936921837, 9.578241429281230, 10.25126860485952, 10.69560944642103, 11.19259025916253, 11.79843819862189, 12.29271610694273, 12.88124400278621, 13.29522545459427, 13.80513481378831, 14.28218060589878, 14.83768763959920

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