Properties

Label 2-46800-1.1-c1-0-141
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 2·11-s + 13-s − 4·17-s + 2·19-s − 2·23-s − 6·29-s − 8·31-s − 6·37-s − 10·41-s − 4·43-s + 9·49-s + 6·53-s + 6·59-s + 2·61-s − 4·67-s − 12·71-s + 2·73-s + 8·77-s − 8·79-s + 12·83-s − 14·89-s − 4·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.603·11-s + 0.277·13-s − 0.970·17-s + 0.458·19-s − 0.417·23-s − 1.11·29-s − 1.43·31-s − 0.986·37-s − 1.56·41-s − 0.609·43-s + 9/7·49-s + 0.824·53-s + 0.781·59-s + 0.256·61-s − 0.488·67-s − 1.42·71-s + 0.234·73-s + 0.911·77-s − 0.900·79-s + 1.31·83-s − 1.48·89-s − 0.419·91-s − 1.01·97-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: \(2\)
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.19844645790894, −14.70765963272777, −13.90887682275256, −13.43683687117501, −13.05924329795892, −12.78408271986622, −11.94180839653617, −11.66046129197330, −10.78173096974000, −10.50691271858265, −9.849154623337592, −9.427615149256613, −8.855774823861581, −8.401592138963335, −7.606612222636670, −6.915330054294903, −6.803594783702234, −5.886199175887072, −5.534447819409623, −4.868113342524662, −3.889546774583695, −3.603228970887885, −2.904748199836488, −2.207729515411550, −1.444346615948420, 0, 0, 1.444346615948420, 2.207729515411550, 2.904748199836488, 3.603228970887885, 3.889546774583695, 4.868113342524662, 5.534447819409623, 5.886199175887072, 6.803594783702234, 6.915330054294903, 7.606612222636670, 8.401592138963335, 8.855774823861581, 9.427615149256613, 9.849154623337592, 10.50691271858265, 10.78173096974000, 11.66046129197330, 11.94180839653617, 12.78408271986622, 13.05924329795892, 13.43683687117501, 13.90887682275256, 14.70765963272777, 15.19844645790894

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