Properties

Label 2-46800-1.1-c1-0-85
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 − 3·11-s + 13-s − 17-s + 6·19-s + 5·23-s + 6·29-s − 2·31-s − 7·37-s − 3·41-s − 8·43-s + 2·47-s + 2·49-s − 53-s − 15·61-s + 12·67-s + 5·71-s + 6·73-s + 9·77-s + 13·79-s − 12·83-s − 89-s − 3·91-s − 17·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.13·7-s − 0.904·11-s + 0.277·13-s − 0.242·17-s + 1.37·19-s + 1.04·23-s + 1.11·29-s − 0.359·31-s − 1.15·37-s − 0.468·41-s − 1.21·43-s + 0.291·47-s + 2/7·49-s − 0.137·53-s − 1.92·61-s + 1.46·67-s + 0.593·71-s + 0.702·73-s + 1.02·77-s + 1.46·79-s − 1.31·83-s − 0.105·89-s − 0.314·91-s − 1.72·97-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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−15.09273121797374, −14.10943347680783, −13.78661486308755, −13.36448769280820, −12.80445091409550, −12.34115516516754, −11.89786891691030, −11.07554164171063, −10.78999117829661, −10.02493324480397, −9.759693927847167, −9.143133588799636, −8.569352708522115, −8.007877058325210, −7.357966451608734, −6.758417879921319, −6.470302300554495, −5.542808305083119, −5.208570064363647, −4.573812190155509, −3.602488045361523, −3.174481431604218, −2.728094441908770, −1.760454023817664, −0.8649557052162369, 0, 0.8649557052162369, 1.760454023817664, 2.728094441908770, 3.174481431604218, 3.602488045361523, 4.573812190155509, 5.208570064363647, 5.542808305083119, 6.470302300554495, 6.758417879921319, 7.357966451608734, 8.007877058325210, 8.569352708522115, 9.143133588799636, 9.759693927847167, 10.02493324480397, 10.78999117829661, 11.07554164171063, 11.89786891691030, 12.34115516516754, 12.80445091409550, 13.36448769280820, 13.78661486308755, 14.10943347680783, 15.09273121797374

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