Properties

Label 2-46800-1.1-c1-0-27
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·7-s + 11-s − 13-s − 5·17-s + 4·19-s + 2·23-s + 3·29-s − 5·31-s + 6·37-s + 8·41-s − 6·43-s + 47-s + 2·49-s + 11·53-s + 3·59-s − 5·61-s + 9·67-s + 8·71-s + 2·73-s − 3·77-s + 4·79-s + 3·83-s − 6·89-s + 3·91-s + 4·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.13·7-s + 0.301·11-s − 0.277·13-s − 1.21·17-s + 0.917·19-s + 0.417·23-s + 0.557·29-s − 0.898·31-s + 0.986·37-s + 1.24·41-s − 0.914·43-s + 0.145·47-s + 2/7·49-s + 1.51·53-s + 0.390·59-s − 0.640·61-s + 1.09·67-s + 0.949·71-s + 0.234·73-s − 0.341·77-s + 0.450·79-s + 0.329·83-s − 0.635·89-s + 0.314·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-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: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.511658647\)
\(L(\frac12)\) \(\approx\) \(1.511658647\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.63347069850657, −13.98442381956777, −13.57252695333705, −12.95413925349683, −12.71195934905742, −12.06395775508238, −11.43114538767837, −11.08898131930743, −10.34931316803237, −9.875751521321037, −9.263033249916397, −9.071978773186502, −8.306573155284095, −7.610462817708167, −7.071538233258405, −6.551832727947177, −6.142780631428461, −5.354201777376250, −4.852563189702134, −3.983852235655148, −3.629547843825064, −2.727901617809251, −2.377385787186084, −1.280346982544511, −0.4590057863552441, 0.4590057863552441, 1.280346982544511, 2.377385787186084, 2.727901617809251, 3.629547843825064, 3.983852235655148, 4.852563189702134, 5.354201777376250, 6.142780631428461, 6.551832727947177, 7.071538233258405, 7.610462817708167, 8.306573155284095, 9.071978773186502, 9.263033249916397, 9.875751521321037, 10.34931316803237, 11.08898131930743, 11.43114538767837, 12.06395775508238, 12.71195934905742, 12.95413925349683, 13.57252695333705, 13.98442381956777, 14.63347069850657

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