Properties

Label 2-46800-1.1-c1-0-3
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  − 3·7-s − 3·11-s + 13-s − 3·17-s + 4·23-s − 5·29-s + 3·31-s − 12·37-s − 2·41-s − 4·43-s + 3·47-s + 2·49-s − 9·53-s + 15·59-s − 3·61-s + 7·67-s − 8·71-s − 16·73-s + 9·77-s − 83-s − 3·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.13·7-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.834·23-s − 0.928·29-s + 0.538·31-s − 1.97·37-s − 0.312·41-s − 0.609·43-s + 0.437·47-s + 2/7·49-s − 1.23·53-s + 1.95·59-s − 0.384·61-s + 0.855·67-s − 0.949·71-s − 1.87·73-s + 1.02·77-s − 0.109·83-s − 0.314·91-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: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5608411497\)
\(L(\frac12)\) \(\approx\) \(0.5608411497\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + T + p T^{2} \)
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

−14.65581598105188, −13.96877228255504, −13.43213934492724, −13.02828248721885, −12.77845737949295, −12.05186022886064, −11.48579288783815, −10.93290484856217, −10.36172731465351, −9.999082686821208, −9.383661161931275, −8.738877099229175, −8.478345190628132, −7.575317172708595, −7.113491188199512, −6.586905619255510, −6.058845987729818, −5.329303965385391, −4.923621137740518, −4.060976691773393, −3.427677301036444, −2.931318653587426, −2.234684611167610, −1.398392584638774, −0.2620265356973018, 0.2620265356973018, 1.398392584638774, 2.234684611167610, 2.931318653587426, 3.427677301036444, 4.060976691773393, 4.923621137740518, 5.329303965385391, 6.058845987729818, 6.586905619255510, 7.113491188199512, 7.575317172708595, 8.478345190628132, 8.738877099229175, 9.383661161931275, 9.999082686821208, 10.36172731465351, 10.93290484856217, 11.48579288783815, 12.05186022886064, 12.77845737949295, 13.02828248721885, 13.43213934492724, 13.96877228255504, 14.65581598105188

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