Properties

Label 2-46800-1.1-c1-0-78
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·7-s + 5·11-s − 13-s − 3·17-s + 4·19-s − 5·23-s + 4·29-s + 7·37-s − 11·41-s − 12·43-s − 6·47-s + 18·49-s + 53-s + 12·59-s − 7·61-s + 4·67-s − 7·71-s + 14·73-s − 25·77-s + 5·79-s + 2·83-s + 3·89-s + 5·91-s + 97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.50·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s − 1.04·23-s + 0.742·29-s + 1.15·37-s − 1.71·41-s − 1.82·43-s − 0.875·47-s + 18/7·49-s + 0.137·53-s + 1.56·59-s − 0.896·61-s + 0.488·67-s − 0.830·71-s + 1.63·73-s − 2.84·77-s + 0.562·79-s + 0.219·83-s + 0.317·89-s + 0.524·91-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 \( 1 + 5 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.96614342286585, −14.27944890921662, −13.65079709971034, −13.44592103215794, −12.82038540310120, −12.21243334562813, −11.81388829270011, −11.48057222888550, −10.55717576411822, −9.962914556370029, −9.659210249186022, −9.307363550887495, −8.576576946187461, −8.125508694337458, −7.116992820012868, −6.804400658749752, −6.361817314447781, −5.932565185162679, −5.064465373915623, −4.393563302684595, −3.553681743923033, −3.457408331523541, −2.579185795613994, −1.790739920181398, −0.8470642118884831, 0, 0.8470642118884831, 1.790739920181398, 2.579185795613994, 3.457408331523541, 3.553681743923033, 4.393563302684595, 5.064465373915623, 5.932565185162679, 6.361817314447781, 6.804400658749752, 7.116992820012868, 8.125508694337458, 8.576576946187461, 9.307363550887495, 9.659210249186022, 9.962914556370029, 10.55717576411822, 11.48057222888550, 11.81388829270011, 12.21243334562813, 12.82038540310120, 13.44592103215794, 13.65079709971034, 14.27944890921662, 14.96614342286585

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