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Properties

Label 2-46800-1.1-c1-0-136
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 + 4·11-s + 13-s − 8·23-s − 8·29-s − 4·31-s − 6·37-s + 12·41-s − 8·43-s − 4·47-s + 9·49-s − 4·59-s − 2·61-s − 8·67-s + 4·71-s + 10·73-s + 16·77-s + 4·79-s + 12·83-s − 12·89-s + 4·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20·11-s + 0.277·13-s − 1.66·23-s − 1.48·29-s − 0.718·31-s − 0.986·37-s + 1.87·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.520·59-s − 0.256·61-s − 0.977·67-s + 0.474·71-s + 1.17·73-s + 1.82·77-s + 0.450·79-s + 1.31·83-s − 1.27·89-s + 0.419·91-s − 1.42·97-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: Trivial
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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.82378013471568, −14.35095337020748, −13.92031766954776, −13.52584787508747, −12.66169011191457, −12.21889758411808, −11.70424224196617, −11.23721039160443, −10.90026236181349, −10.24129702479240, −9.474531056919516, −9.159329718058599, −8.481576756445774, −7.958521508609361, −7.586731674383915, −6.868093066649215, −6.240162066000085, −5.647037305938179, −5.126572140149431, −4.361132013043644, −3.944241280979255, −3.380821592519285, −2.203478502555485, −1.758384645605944, −1.199755940553330, 0, 1.199755940553330, 1.758384645605944, 2.203478502555485, 3.380821592519285, 3.944241280979255, 4.361132013043644, 5.126572140149431, 5.647037305938179, 6.240162066000085, 6.868093066649215, 7.586731674383915, 7.958521508609361, 8.481576756445774, 9.159329718058599, 9.474531056919516, 10.24129702479240, 10.90026236181349, 11.23721039160443, 11.70424224196617, 12.21889758411808, 12.66169011191457, 13.52584787508747, 13.92031766954776, 14.35095337020748, 14.82378013471568

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