Properties

Label 2-4560-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 3.20·7-s + 9-s − 5.20·11-s + 5.45·13-s − 15-s + 4·17-s − 19-s + 3.20·21-s + 8.65·23-s + 25-s − 27-s − 3.20·29-s + 2·31-s + 5.20·33-s − 3.20·35-s − 11.8·37-s − 5.45·39-s − 9.85·41-s + 3.45·43-s + 45-s − 8.65·47-s + 3.25·49-s − 4·51-s − 10.6·53-s − 5.20·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.21·7-s + 0.333·9-s − 1.56·11-s + 1.51·13-s − 0.258·15-s + 0.970·17-s − 0.229·19-s + 0.698·21-s + 1.80·23-s + 0.200·25-s − 0.192·27-s − 0.594·29-s + 0.359·31-s + 0.905·33-s − 0.541·35-s − 1.94·37-s − 0.873·39-s − 1.53·41-s + 0.526·43-s + 0.149·45-s − 1.26·47-s + 0.464·49-s − 0.560·51-s − 1.46·53-s − 0.701·55-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: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.237031170\)
\(L(\frac12)\) \(\approx\) \(1.237031170\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.20T + 7T^{2} \)
11 \( 1 + 5.20T + 11T^{2} \)
13 \( 1 - 5.45T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
23 \( 1 - 8.65T + 23T^{2} \)
29 \( 1 + 3.20T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + 9.85T + 41T^{2} \)
43 \( 1 - 3.45T + 43T^{2} \)
47 \( 1 + 8.65T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 3.74T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 6.90T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 - 8.40T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 0.142T + 97T^{2} \)
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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

−8.405277707179742552915524993825, −7.48603282466043085512740391034, −6.70805336271148444250210561791, −6.17428763166814308738176458374, −5.37354205278993409376935154125, −4.93729325026524221985319285821, −3.41701869388460455000460819819, −3.21210915509482836846883328993, −1.84703216240304355420527762186, −0.63260094896920489915215132055, 0.63260094896920489915215132055, 1.84703216240304355420527762186, 3.21210915509482836846883328993, 3.41701869388460455000460819819, 4.93729325026524221985319285821, 5.37354205278993409376935154125, 6.17428763166814308738176458374, 6.70805336271148444250210561791, 7.48603282466043085512740391034, 8.405277707179742552915524993825

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