Properties

Label 2-46800-1.1-c1-0-65
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  − 4·7-s + 2·11-s + 13-s + 2·17-s + 6·19-s + 6·23-s − 2·29-s + 10·31-s + 2·37-s + 6·41-s + 10·43-s − 4·47-s + 9·49-s + 2·53-s + 6·59-s + 2·61-s − 4·67-s + 6·71-s + 6·73-s − 8·77-s + 12·79-s + 16·83-s − 2·89-s − 4·91-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.603·11-s + 0.277·13-s + 0.485·17-s + 1.37·19-s + 1.25·23-s − 0.371·29-s + 1.79·31-s + 0.328·37-s + 0.937·41-s + 1.52·43-s − 0.583·47-s + 9/7·49-s + 0.274·53-s + 0.781·59-s + 0.256·61-s − 0.488·67-s + 0.712·71-s + 0.702·73-s − 0.911·77-s + 1.35·79-s + 1.75·83-s − 0.211·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(2.779693579\)
\(L(\frac12)\) \(\approx\) \(2.779693579\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−14.59124060964456, −13.97928308574504, −13.56430096754948, −13.07427821308572, −12.55729813126915, −12.08133515137741, −11.57251268915096, −10.98163673073443, −10.36600374469497, −9.781313690191153, −9.366200695391879, −9.088100644230147, −8.267811101713356, −7.596681498515287, −7.136322987599784, −6.423982895480456, −6.189583568931414, −5.430499435777477, −4.840529060325049, −3.984330429578789, −3.465718418053423, −2.953488803322256, −2.333735242322442, −1.036188489426497, −0.7529727091431540, 0.7529727091431540, 1.036188489426497, 2.333735242322442, 2.953488803322256, 3.465718418053423, 3.984330429578789, 4.840529060325049, 5.430499435777477, 6.189583568931414, 6.423982895480456, 7.136322987599784, 7.596681498515287, 8.267811101713356, 9.088100644230147, 9.366200695391879, 9.781313690191153, 10.36600374469497, 10.98163673073443, 11.57251268915096, 12.08133515137741, 12.55729813126915, 13.07427821308572, 13.56430096754948, 13.97928308574504, 14.59124060964456

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