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Properties

Label 1-1020-1020.1019-r0-0-0
Degree $1$
Conductor $1020$
Sign $1$
Analytic cond. $4.73686$
Root an. cond. $4.73686$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 11-s − 13-s − 19-s + 23-s + 29-s + 31-s + 37-s + 41-s + 43-s − 47-s + 49-s + 53-s + 59-s − 61-s + 67-s − 71-s + 73-s + 77-s + 79-s − 83-s − 89-s + 91-s + 97-s + ⋯
L(s)  = 1  − 7-s − 11-s − 13-s − 19-s + 23-s + 29-s + 31-s + 37-s + 41-s + 43-s − 47-s + 49-s + 53-s + 59-s − 61-s + 67-s − 71-s + 73-s + 77-s + 79-s − 83-s − 89-s + 91-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $1$
Analytic conductor: \(4.73686\)
Root analytic conductor: \(4.73686\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1020} (1019, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1020,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.019852647\)
\(L(\frac12)\) \(\approx\) \(1.019852647\)
\(L(1)\) \(\approx\) \(0.8678862255\)
\(L(1)\) \(\approx\) \(0.8678862255\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.435396823030357949920056037793, −21.05591420187654466978296271867, −19.79323219788299819669411768665, −19.38385024488726916503718026960, −18.6028092316135381760460227382, −17.64373664848921508888609464179, −16.888526002872300623501131381679, −16.09597086319482635254523235129, −15.361338402839808934700444160705, −14.60761198833890646422531928832, −13.53614757891902870619102116572, −12.82994538460614355393785737100, −12.289396254917716864183230644954, −11.11729284624291135740251455689, −10.27197472857869412167487366675, −9.660543010320688880861994098085, −8.68780993996973916923702262051, −7.74717622358740240067473227601, −6.87617714484121166936165210670, −6.075819555824401702102880966373, −5.0475120865529955947277677078, −4.19821236635143829245308907688, −2.89403483972267077633535337397, −2.426799448940788830750715673752, −0.69392957771563392248788015509, 0.69392957771563392248788015509, 2.426799448940788830750715673752, 2.89403483972267077633535337397, 4.19821236635143829245308907688, 5.0475120865529955947277677078, 6.075819555824401702102880966373, 6.87617714484121166936165210670, 7.74717622358740240067473227601, 8.68780993996973916923702262051, 9.660543010320688880861994098085, 10.27197472857869412167487366675, 11.11729284624291135740251455689, 12.289396254917716864183230644954, 12.82994538460614355393785737100, 13.53614757891902870619102116572, 14.60761198833890646422531928832, 15.361338402839808934700444160705, 16.09597086319482635254523235129, 16.888526002872300623501131381679, 17.64373664848921508888609464179, 18.6028092316135381760460227382, 19.38385024488726916503718026960, 19.79323219788299819669411768665, 21.05591420187654466978296271867, 21.435396823030357949920056037793

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