Properties

Label 2-188760-1.1-c1-0-27
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s − 13-s + 15-s − 7·17-s + 4·19-s − 21-s + 25-s − 27-s − 5·29-s + 2·31-s − 35-s − 8·37-s + 39-s + 10·41-s + 43-s − 45-s − 8·47-s − 6·49-s + 7·51-s − 4·57-s − 12·59-s − 10·61-s + 63-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 1.69·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 0.359·31-s − 0.169·35-s − 1.31·37-s + 0.160·39-s + 1.56·41-s + 0.152·43-s − 0.149·45-s − 1.16·47-s − 6/7·49-s + 0.980·51-s − 0.529·57-s − 1.56·59-s − 1.28·61-s + 0.125·63-s + 0.124·65-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: $\chi_{188760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.29272772644821, −12.76941822108351, −12.47628868782773, −11.76684878640633, −11.48616115187964, −11.05434405683054, −10.75662831480143, −10.03456557678634, −9.636855529572699, −9.023010606221947, −8.659248784772919, −8.074018531063947, −7.380043674138597, −7.248818645038924, −6.632146911949418, −5.952588315719791, −5.669152972504111, −4.862153044749258, −4.491511005616159, −4.212902092202130, −3.195983382347173, −2.983636626656155, −1.871801605626213, −1.673005540108721, −0.6311512002675629, 0, 0.6311512002675629, 1.673005540108721, 1.871801605626213, 2.983636626656155, 3.195983382347173, 4.212902092202130, 4.491511005616159, 4.862153044749258, 5.669152972504111, 5.952588315719791, 6.632146911949418, 7.248818645038924, 7.380043674138597, 8.074018531063947, 8.659248784772919, 9.023010606221947, 9.636855529572699, 10.03456557678634, 10.75662831480143, 11.05434405683054, 11.48616115187964, 11.76684878640633, 12.47628868782773, 12.76941822108351, 13.29272772644821

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