Properties

Label 2-46800-1.1-c1-0-12
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  − 13-s − 6·17-s − 4·19-s + 2·29-s + 2·37-s + 2·41-s − 4·43-s − 4·47-s − 7·49-s − 10·53-s + 8·59-s − 2·61-s − 4·67-s − 12·71-s + 6·73-s + 16·83-s + 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.277·13-s − 1.45·17-s − 0.917·19-s + 0.371·29-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.583·47-s − 49-s − 1.37·53-s + 1.04·59-s − 0.256·61-s − 0.488·67-s − 1.42·71-s + 0.702·73-s + 1.75·83-s + 1.05·89-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.071719982\)
\(L(\frac12)\) \(\approx\) \(1.071719982\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.70721325601361, −14.11676604754833, −13.43910268433716, −13.11136568361687, −12.63859357264529, −12.00596123613498, −11.44485604542293, −11.01484412958511, −10.45839166432566, −9.978593309638912, −9.186626111614066, −8.986630559943733, −8.164912114463387, −7.871752444820437, −7.030958937131245, −6.465904220535080, −6.238174739209557, −5.274178971866294, −4.727536596771065, −4.270296936584620, −3.538632722270980, −2.768854493267974, −2.159872171848166, −1.488328825348248, −0.3566354593083553, 0.3566354593083553, 1.488328825348248, 2.159872171848166, 2.768854493267974, 3.538632722270980, 4.270296936584620, 4.727536596771065, 5.274178971866294, 6.238174739209557, 6.465904220535080, 7.030958937131245, 7.871752444820437, 8.164912114463387, 8.986630559943733, 9.186626111614066, 9.978593309638912, 10.45839166432566, 11.01484412958511, 11.44485604542293, 12.00596123613498, 12.63859357264529, 13.11136568361687, 13.43910268433716, 14.11676604754833, 14.70721325601361

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