Properties

Label 2-31200-1.1-c1-0-51
Degree $2$
Conductor $31200$
Sign $-1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
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 + 3·11-s + 13-s + 3·17-s − 6·19-s − 21-s + 23-s + 27-s + 8·29-s + 4·31-s + 3·33-s − 5·37-s + 39-s − 5·41-s − 6·43-s + 8·47-s − 6·49-s + 3·51-s − 9·53-s − 6·57-s + 4·59-s − 11·61-s − 63-s − 16·67-s + 69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.727·17-s − 1.37·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 1.48·29-s + 0.718·31-s + 0.522·33-s − 0.821·37-s + 0.160·39-s − 0.780·41-s − 0.914·43-s + 1.16·47-s − 6/7·49-s + 0.420·51-s − 1.23·53-s − 0.794·57-s + 0.520·59-s − 1.40·61-s − 0.125·63-s − 1.95·67-s + 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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

−15.20452496459086, −14.89405287764761, −14.24155799651224, −13.82695287875504, −13.34366867747285, −12.71416423794825, −12.10084490984116, −11.89518696904166, −11.00094080118796, −10.31320814813421, −10.16248511399649, −9.178157183872183, −9.021894547686681, −8.197428703510875, −7.973438733268510, −6.922642088062120, −6.622267346731879, −6.099968391010571, −5.256023804456283, −4.452420524770176, −4.062075209330568, −3.170835335755333, −2.847680785313527, −1.758071000136861, −1.224469032840309, 0, 1.224469032840309, 1.758071000136861, 2.847680785313527, 3.170835335755333, 4.062075209330568, 4.452420524770176, 5.256023804456283, 6.099968391010571, 6.622267346731879, 6.922642088062120, 7.973438733268510, 8.197428703510875, 9.021894547686681, 9.178157183872183, 10.16248511399649, 10.31320814813421, 11.00094080118796, 11.89518696904166, 12.10084490984116, 12.71416423794825, 13.34366867747285, 13.82695287875504, 14.24155799651224, 14.89405287764761, 15.20452496459086

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