Properties

Label 2-46800-1.1-c1-0-46
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 − 13-s + 6·17-s + 4·19-s + 6·23-s − 6·29-s + 10·31-s + 10·37-s − 6·41-s − 4·43-s + 12·47-s + 9·49-s − 12·53-s + 12·59-s − 10·61-s + 14·67-s + 16·73-s − 8·79-s + 12·83-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 + 1.45·17-s + 0.917·19-s + 1.25·23-s − 1.11·29-s + 1.79·31-s + 1.64·37-s − 0.937·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s − 1.64·53-s + 1.56·59-s − 1.28·61-s + 1.71·67-s + 1.87·73-s − 0.900·79-s + 1.31·83-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: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.225092627\)
\(L(\frac12)\) \(\approx\) \(2.225092627\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−14.56545879202336, −14.03210808491390, −13.54830153408616, −12.96913427374230, −12.65304351924641, −12.06293918535692, −11.59822920106278, −10.98909441975665, −10.23621220118068, −9.861815499610495, −9.474794372477171, −9.031027727876019, −8.163498140900817, −7.669169778086498, −7.149277143164292, −6.486828373733404, −6.113833767752095, −5.344111069315757, −4.955985569660868, −3.992366496039851, −3.378645117800980, −2.998620902080571, −2.334433868122715, −1.136657557431172, −0.6147219417049466, 0.6147219417049466, 1.136657557431172, 2.334433868122715, 2.998620902080571, 3.378645117800980, 3.992366496039851, 4.955985569660868, 5.344111069315757, 6.113833767752095, 6.486828373733404, 7.149277143164292, 7.669169778086498, 8.163498140900817, 9.031027727876019, 9.474794372477171, 9.861815499610495, 10.23621220118068, 10.98909441975665, 11.59822920106278, 12.06293918535692, 12.65304351924641, 12.96913427374230, 13.54830153408616, 14.03210808491390, 14.56545879202336

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