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Properties

Label 2-65520-1.1-c1-0-14
Degree $2$
Conductor $65520$
Sign $1$
Analytic cond. $523.179$
Root an. cond. $22.8731$
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  − 5-s − 7-s − 2·11-s − 13-s + 4·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s + 2·31-s + 35-s + 6·37-s + 2·43-s − 8·47-s + 49-s + 12·53-s + 2·55-s − 4·59-s − 2·61-s + 65-s + 16·67-s − 12·71-s + 6·73-s + 2·77-s − 4·83-s − 4·85-s + 91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.603·11-s − 0.277·13-s + 0.970·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 0.359·31-s + 0.169·35-s + 0.986·37-s + 0.304·43-s − 1.16·47-s + 1/7·49-s + 1.64·53-s + 0.269·55-s − 0.520·59-s − 0.256·61-s + 0.124·65-s + 1.95·67-s − 1.42·71-s + 0.702·73-s + 0.227·77-s − 0.439·83-s − 0.433·85-s + 0.104·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65520\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(523.179\)
Root analytic conductor: \(22.8731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 65520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.662191261\)
\(L(\frac12)\) \(\approx\) \(1.662191261\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.35016655226033, −13.65730952491522, −13.05591070294614, −12.77800130985931, −12.28125877507792, −11.65994776437369, −11.26052408825902, −10.59037182383531, −10.18885916711372, −9.718646846387702, −9.078144622290285, −8.482296973195662, −8.074179472789975, −7.434060118356167, −7.061863565998001, −6.295117399415035, −5.909756882472367, −5.072347584417981, −4.743341987978464, −3.963806708479621, −3.398361297142069, −2.753197285401091, −2.224392560257461, −1.178014390552003, −0.4741691106247661, 0.4741691106247661, 1.178014390552003, 2.224392560257461, 2.753197285401091, 3.398361297142069, 3.963806708479621, 4.743341987978464, 5.072347584417981, 5.909756882472367, 6.295117399415035, 7.061863565998001, 7.434060118356167, 8.074179472789975, 8.482296973195662, 9.078144622290285, 9.718646846387702, 10.18885916711372, 10.59037182383531, 11.26052408825902, 11.65994776437369, 12.28125877507792, 12.77800130985931, 13.05591070294614, 13.65730952491522, 14.35016655226033

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