Properties

Label 2-4560-1.1-c1-0-38
Degree $2$
Conductor $4560$
Sign $-1$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 5.23·7-s + 9-s + 1.23·11-s + 0.763·13-s + 15-s + 4.47·17-s − 19-s + 5.23·21-s − 2.47·23-s + 25-s − 27-s + 0.763·29-s + 8.94·31-s − 1.23·33-s + 5.23·35-s + 3.23·37-s − 0.763·39-s − 9.70·41-s − 5.23·43-s − 45-s − 2.47·47-s + 20.4·49-s − 4.47·51-s − 0.472·53-s − 1.23·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.97·7-s + 0.333·9-s + 0.372·11-s + 0.211·13-s + 0.258·15-s + 1.08·17-s − 0.229·19-s + 1.14·21-s − 0.515·23-s + 0.200·25-s − 0.192·27-s + 0.141·29-s + 1.60·31-s − 0.215·33-s + 0.885·35-s + 0.532·37-s − 0.122·39-s − 1.51·41-s − 0.798·43-s − 0.149·45-s − 0.360·47-s + 2.91·49-s − 0.626·51-s − 0.0648·53-s − 0.166·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
19 \( 1 + T \)
good7 \( 1 + 5.23T + 7T^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 - 0.763T + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 - 0.763T + 29T^{2} \)
31 \( 1 - 8.94T + 31T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
41 \( 1 + 9.70T + 41T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 + 2.47T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 1.70T + 89T^{2} \)
97 \( 1 + 4.76T + 97T^{2} \)
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

−7.979039525906234825297358163718, −6.93004866044176360801038118325, −6.57112611864092170321474058583, −5.93264050198987387368779907693, −5.09608758793791000648363443646, −3.99777403005525344189530428309, −3.47625043537095783223411719979, −2.62897139328910396580103982469, −1.07256167169013163594337395325, 0, 1.07256167169013163594337395325, 2.62897139328910396580103982469, 3.47625043537095783223411719979, 3.99777403005525344189530428309, 5.09608758793791000648363443646, 5.93264050198987387368779907693, 6.57112611864092170321474058583, 6.93004866044176360801038118325, 7.979039525906234825297358163718

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