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Properties

Label 2-43560-1.1-c1-0-28
Degree $2$
Conductor $43560$
Sign $1$
Analytic cond. $347.828$
Root an. cond. $18.6501$
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  − 5-s + 3·7-s + 5·13-s + 5·17-s − 19-s + 4·23-s + 25-s − 5·29-s + 2·31-s − 3·35-s + 3·37-s + 2·41-s − 2·43-s − 6·47-s + 2·49-s − 4·59-s − 8·61-s − 5·65-s + 6·67-s − 3·71-s + 10·73-s + 10·79-s + 9·83-s − 5·85-s + 12·89-s + 15·91-s + 95-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s + 1.38·13-s + 1.21·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s − 0.928·29-s + 0.359·31-s − 0.507·35-s + 0.493·37-s + 0.312·41-s − 0.304·43-s − 0.875·47-s + 2/7·49-s − 0.520·59-s − 1.02·61-s − 0.620·65-s + 0.733·67-s − 0.356·71-s + 1.17·73-s + 1.12·79-s + 0.987·83-s − 0.542·85-s + 1.27·89-s + 1.57·91-s + 0.102·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43560\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(347.828\)
Root analytic conductor: \(18.6501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 43560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.211674450\)
\(L(\frac12)\) \(\approx\) \(3.211674450\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 \)
good7 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.80574225811537, −14.20766128979765, −13.66967082506970, −13.20145617244751, −12.55496316058252, −12.07644587468465, −11.42374091082670, −11.09089855014953, −10.73662468838260, −10.01642932760774, −9.319765253631944, −8.830815886788859, −8.163769460564879, −7.884038677915675, −7.381059294212245, −6.520494835469892, −6.077454744799666, −5.279867742127422, −4.912625969531790, −4.147697442847494, −3.560101840058141, −3.034103546728676, −1.998205386242448, −1.352258205748541, −0.7098556421076664, 0.7098556421076664, 1.352258205748541, 1.998205386242448, 3.034103546728676, 3.560101840058141, 4.147697442847494, 4.912625969531790, 5.279867742127422, 6.077454744799666, 6.520494835469892, 7.381059294212245, 7.884038677915675, 8.163769460564879, 8.830815886788859, 9.319765253631944, 10.01642932760774, 10.73662468838260, 11.09089855014953, 11.42374091082670, 12.07644587468465, 12.55496316058252, 13.20145617244751, 13.66967082506970, 14.20766128979765, 14.80574225811537

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