Properties

Label 2-188760-1.1-c1-0-47
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 2·7-s + 9-s − 13-s + 15-s − 3·17-s + 4·19-s − 2·21-s + 5·23-s + 25-s − 27-s − 9·31-s − 2·35-s + 2·37-s + 39-s − 4·43-s − 45-s + 13·47-s − 3·49-s + 3·51-s − 53-s − 4·57-s − 6·59-s − 13·61-s + 2·63-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.917·19-s − 0.436·21-s + 1.04·23-s + 1/5·25-s − 0.192·27-s − 1.61·31-s − 0.338·35-s + 0.328·37-s + 0.160·39-s − 0.609·43-s − 0.149·45-s + 1.89·47-s − 3/7·49-s + 0.420·51-s − 0.137·53-s − 0.529·57-s − 0.781·59-s − 1.66·61-s + 0.251·63-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.42732660391496, −12.71457754126362, −12.35101360296789, −11.99147457208333, −11.26829998915621, −11.09174631649180, −10.81483075219380, −10.11411683923851, −9.506783484996466, −9.105466803472259, −8.643322166872618, −7.991560528314263, −7.523004398615242, −7.150620841135064, −6.706898313388040, −5.972462443272503, −5.472909935591810, −5.027622349048759, −4.547499987987315, −4.065807383805731, −3.369175733781667, −2.826997537882222, −2.030874736729245, −1.461917440098309, −0.7739695191948948, 0, 0.7739695191948948, 1.461917440098309, 2.030874736729245, 2.826997537882222, 3.369175733781667, 4.065807383805731, 4.547499987987315, 5.027622349048759, 5.472909935591810, 5.972462443272503, 6.706898313388040, 7.150620841135064, 7.523004398615242, 7.991560528314263, 8.643322166872618, 9.105466803472259, 9.506783484996466, 10.11411683923851, 10.81483075219380, 11.09174631649180, 11.26829998915621, 11.99147457208333, 12.35101360296789, 12.71457754126362, 13.42732660391496

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