Properties

Label 2-188760-1.1-c1-0-10
Degree $2$
Conductor $188760$
Sign $1$
Analytic cond. $1507.25$
Root an. cond. $38.8233$
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  − 3-s + 5-s + 9-s − 13-s − 15-s + 2·17-s + 4·19-s + 4·23-s + 25-s − 27-s − 4·31-s − 8·37-s + 39-s + 6·41-s + 8·43-s + 45-s + 2·47-s − 7·49-s − 2·51-s − 6·53-s − 4·57-s − 4·59-s − 2·61-s − 65-s + 2·67-s − 4·69-s − 14·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.718·31-s − 1.31·37-s + 0.160·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 0.291·47-s − 49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.124·65-s + 0.244·67-s − 0.481·69-s − 1.66·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(188760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1507.25\)
Root analytic conductor: \(38.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.850200285\)
\(L(\frac12)\) \(\approx\) \(1.850200285\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.97756912091696, −12.62415894529745, −12.26201656893686, −11.66354765380965, −11.24833833711276, −10.75840204865461, −10.37771047121812, −9.722118587572908, −9.497006982176900, −8.883379626865363, −8.460228155215401, −7.555613144467101, −7.411137763817965, −6.905497299093699, −6.209336800297290, −5.718795701817937, −5.449110055829439, −4.734026912765020, −4.431090459365508, −3.531037108838191, −3.114803710862487, −2.489802398074868, −1.647824593450485, −1.241062171948573, −0.4189660486956664, 0.4189660486956664, 1.241062171948573, 1.647824593450485, 2.489802398074868, 3.114803710862487, 3.531037108838191, 4.431090459365508, 4.734026912765020, 5.449110055829439, 5.718795701817937, 6.209336800297290, 6.905497299093699, 7.411137763817965, 7.555613144467101, 8.460228155215401, 8.883379626865363, 9.497006982176900, 9.722118587572908, 10.37771047121812, 10.75840204865461, 11.24833833711276, 11.66354765380965, 12.26201656893686, 12.62415894529745, 12.97756912091696

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