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Properties

Label 2-10800-1.1-c1-0-29
Degree $2$
Conductor $10800$
Sign $1$
Analytic cond. $86.2384$
Root an. cond. $9.28646$
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  + 7-s + 2·13-s + 7·19-s + 7·31-s − 37-s − 5·43-s − 6·49-s − 61-s + 16·67-s + 17·73-s + 13·79-s + 2·91-s − 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.377·7-s + 0.554·13-s + 1.60·19-s + 1.25·31-s − 0.164·37-s − 0.762·43-s − 6/7·49-s − 0.128·61-s + 1.95·67-s + 1.98·73-s + 1.46·79-s + 0.209·91-s − 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10800\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(86.2384\)
Root analytic conductor: \(9.28646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.491760514\)
\(L(\frac12)\) \(\approx\) \(2.491760514\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−16.46973285495099, −15.90817496735779, −15.47949820809805, −14.84665889102824, −14.09809830841093, −13.77172383648524, −13.20961327683561, −12.39393926974096, −11.91211672535954, −11.29053026198035, −10.85262519935651, −9.963097319565227, −9.599343928344968, −8.846541451193521, −8.056212759735632, −7.815019346317432, −6.781951919046580, −6.408978018260062, −5.341687802401857, −5.080489326979188, −4.078048123789853, −3.396320928259907, −2.621566552219375, −1.591255322818495, −0.7806578258863872, 0.7806578258863872, 1.591255322818495, 2.621566552219375, 3.396320928259907, 4.078048123789853, 5.080489326979188, 5.341687802401857, 6.408978018260062, 6.781951919046580, 7.815019346317432, 8.056212759735632, 8.846541451193521, 9.599343928344968, 9.963097319565227, 10.85262519935651, 11.29053026198035, 11.91211672535954, 12.39393926974096, 13.20961327683561, 13.77172383648524, 14.09809830841093, 14.84665889102824, 15.47949820809805, 15.90817496735779, 16.46973285495099

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