Properties

Label 2-46800-1.1-c1-0-68
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  − 3·7-s − 5·11-s − 13-s + 5·17-s − 2·19-s + 23-s − 10·29-s + 2·31-s + 3·37-s + 9·41-s − 4·43-s − 10·47-s + 2·49-s + 9·53-s − 11·61-s − 4·67-s + 15·71-s − 6·73-s + 15·77-s + 11·79-s − 8·83-s + 11·89-s + 3·91-s + 9·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.13·7-s − 1.50·11-s − 0.277·13-s + 1.21·17-s − 0.458·19-s + 0.208·23-s − 1.85·29-s + 0.359·31-s + 0.493·37-s + 1.40·41-s − 0.609·43-s − 1.45·47-s + 2/7·49-s + 1.23·53-s − 1.40·61-s − 0.488·67-s + 1.78·71-s − 0.702·73-s + 1.70·77-s + 1.23·79-s − 0.878·83-s + 1.16·89-s + 0.314·91-s + 0.913·97-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−14.83665536456322, −14.48366318962131, −13.66434312992616, −13.11259030427328, −12.97571614946843, −12.40392037999853, −11.87021942830415, −11.09960255545260, −10.71516416774044, −10.05799295385167, −9.726123189276232, −9.260701911377926, −8.495231897206263, −7.873174782342494, −7.484067757219040, −6.962367715123653, −5.981800632148425, −5.932711224475636, −5.085134449332837, −4.594226880566359, −3.574944055720267, −3.286095347068071, −2.544484630001115, −1.928531858363539, −0.7605072245259315, 0, 0.7605072245259315, 1.928531858363539, 2.544484630001115, 3.286095347068071, 3.574944055720267, 4.594226880566359, 5.085134449332837, 5.932711224475636, 5.981800632148425, 6.962367715123653, 7.484067757219040, 7.873174782342494, 8.495231897206263, 9.260701911377926, 9.726123189276232, 10.05799295385167, 10.71516416774044, 11.09960255545260, 11.87021942830415, 12.40392037999853, 12.97571614946843, 13.11259030427328, 13.66434312992616, 14.48366318962131, 14.83665536456322

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