Properties

Label 2-46800-1.1-c1-0-71
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  − 4·7-s − 13-s − 5·19-s − 3·29-s + 4·31-s + 7·37-s − 3·41-s + 2·43-s − 9·47-s + 9·49-s + 9·53-s + 6·59-s + 8·61-s + 5·67-s − 3·71-s + 4·73-s − 11·79-s + 6·83-s − 6·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.277·13-s − 1.14·19-s − 0.557·29-s + 0.718·31-s + 1.15·37-s − 0.468·41-s + 0.304·43-s − 1.31·47-s + 9/7·49-s + 1.23·53-s + 0.781·59-s + 1.02·61-s + 0.610·67-s − 0.356·71-s + 0.468·73-s − 1.23·79-s + 0.658·83-s − 0.635·89-s + 0.419·91-s − 0.812·97-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.94480932189723, −14.41109966102428, −13.68929791795053, −13.19101297847445, −12.87467320084595, −12.43972881797652, −11.77302830932168, −11.28808453813124, −10.59889327288989, −10.10040185053372, −9.652002833451386, −9.260280138147368, −8.470328842861699, −8.141231800466415, −7.272861487018973, −6.729035236527830, −6.430168495017939, −5.733192176795932, −5.207860252309029, −4.230286513598577, −3.962978592467677, −3.081278787771592, −2.631884671118120, −1.880503850261917, −0.7787349513776047, 0, 0.7787349513776047, 1.880503850261917, 2.631884671118120, 3.081278787771592, 3.962978592467677, 4.230286513598577, 5.207860252309029, 5.733192176795932, 6.430168495017939, 6.729035236527830, 7.272861487018973, 8.141231800466415, 8.470328842861699, 9.260280138147368, 9.652002833451386, 10.10040185053372, 10.59889327288989, 11.28808453813124, 11.77302830932168, 12.43972881797652, 12.87467320084595, 13.19101297847445, 13.68929791795053, 14.41109966102428, 14.94480932189723

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