Properties

Label 2-46800-1.1-c1-0-99
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  + 7-s − 3·11-s + 13-s − 3·17-s + 4·19-s − 6·23-s − 3·29-s + 31-s + 2·37-s + 10·43-s − 3·47-s − 6·49-s + 3·53-s + 3·59-s + 5·61-s + 7·67-s + 2·73-s − 3·77-s + 10·79-s − 15·83-s + 91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s − 1.25·23-s − 0.557·29-s + 0.179·31-s + 0.328·37-s + 1.52·43-s − 0.437·47-s − 6/7·49-s + 0.412·53-s + 0.390·59-s + 0.640·61-s + 0.855·67-s + 0.234·73-s − 0.341·77-s + 1.12·79-s − 1.64·83-s + 0.104·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.84388116583925, −14.25921414348066, −13.87508981727475, −13.30095344972607, −12.87364830671265, −12.31109256157207, −11.66536691989439, −11.26600201598810, −10.74843861807492, −10.21556530753333, −9.586286053140568, −9.218972659070211, −8.323105548690419, −8.088193770328366, −7.507383229834645, −6.891164096655293, −6.254539248260177, −5.547545117800584, −5.248005755596706, −4.389124699284708, −3.959293576842750, −3.136322949429250, −2.444556252746293, −1.868063036160098, −0.9427492501812985, 0, 0.9427492501812985, 1.868063036160098, 2.444556252746293, 3.136322949429250, 3.959293576842750, 4.389124699284708, 5.248005755596706, 5.547545117800584, 6.254539248260177, 6.891164096655293, 7.507383229834645, 8.088193770328366, 8.323105548690419, 9.218972659070211, 9.586286053140568, 10.21556530753333, 10.74843861807492, 11.26600201598810, 11.66536691989439, 12.31109256157207, 12.87364830671265, 13.30095344972607, 13.87508981727475, 14.25921414348066, 14.84388116583925

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