Properties

Label 2-46800-1.1-c1-0-62
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  + 7-s − 3·11-s + 13-s + 17-s + 8·19-s + 4·23-s + 7·29-s − 31-s + 4·37-s + 6·41-s + 12·43-s + 3·47-s − 6·49-s − 5·53-s − 9·59-s + 5·61-s + 11·67-s + 8·71-s − 3·77-s + 8·79-s + 7·83-s + 8·89-s + 91-s + 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.904·11-s + 0.277·13-s + 0.242·17-s + 1.83·19-s + 0.834·23-s + 1.29·29-s − 0.179·31-s + 0.657·37-s + 0.937·41-s + 1.82·43-s + 0.437·47-s − 6/7·49-s − 0.686·53-s − 1.17·59-s + 0.640·61-s + 1.34·67-s + 0.949·71-s − 0.341·77-s + 0.900·79-s + 0.768·83-s + 0.847·89-s + 0.104·91-s + 0.609·97-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: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.147517737\)
\(L(\frac12)\) \(\approx\) \(3.147517737\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.46595804053177, −14.02154442220595, −13.76759563773794, −12.89104709292285, −12.70445888882462, −11.99311783131051, −11.47089371440272, −10.94067298599115, −10.57145890335878, −9.862094798857361, −9.348402655924650, −8.944196491387996, −7.982600038069065, −7.851483186516089, −7.298606450402560, −6.523426012998334, −5.972310506262881, −5.168438967506147, −5.034646259579428, −4.194641935483963, −3.420627182453691, −2.838333988672516, −2.276087781008016, −1.176120481189265, −0.7343524184532013, 0.7343524184532013, 1.176120481189265, 2.276087781008016, 2.838333988672516, 3.420627182453691, 4.194641935483963, 5.034646259579428, 5.168438967506147, 5.972310506262881, 6.523426012998334, 7.298606450402560, 7.851483186516089, 7.982600038069065, 8.944196491387996, 9.348402655924650, 9.862094798857361, 10.57145890335878, 10.94067298599115, 11.47089371440272, 11.99311783131051, 12.70445888882462, 12.89104709292285, 13.76759563773794, 14.02154442220595, 14.46595804053177

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