Properties

Label 2-8020-1.1-c1-0-23
Degree $2$
Conductor $8020$
Sign $1$
Analytic cond. $64.0400$
Root an. cond. $8.00250$
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  − 3.05·3-s + 5-s + 2.86·7-s + 6.31·9-s − 2.92·11-s + 2.59·13-s − 3.05·15-s − 3.00·17-s − 6.44·19-s − 8.73·21-s − 6.65·23-s + 25-s − 10.1·27-s + 0.745·29-s − 6.77·31-s + 8.92·33-s + 2.86·35-s + 3.42·37-s − 7.90·39-s + 10.4·41-s − 9.67·43-s + 6.31·45-s + 1.71·47-s + 1.18·49-s + 9.17·51-s − 1.95·53-s − 2.92·55-s + ⋯
L(s)  = 1  − 1.76·3-s + 0.447·5-s + 1.08·7-s + 2.10·9-s − 0.881·11-s + 0.718·13-s − 0.788·15-s − 0.729·17-s − 1.47·19-s − 1.90·21-s − 1.38·23-s + 0.200·25-s − 1.94·27-s + 0.138·29-s − 1.21·31-s + 1.55·33-s + 0.483·35-s + 0.563·37-s − 1.26·39-s + 1.63·41-s − 1.47·43-s + 0.941·45-s + 0.249·47-s + 0.168·49-s + 1.28·51-s − 0.267·53-s − 0.394·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
Sign: $1$
Analytic conductor: \(64.0400\)
Root analytic conductor: \(8.00250\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9002714710\)
\(L(\frac12)\) \(\approx\) \(0.9002714710\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
401 \( 1 - T \)
good3 \( 1 + 3.05T + 3T^{2} \)
7 \( 1 - 2.86T + 7T^{2} \)
11 \( 1 + 2.92T + 11T^{2} \)
13 \( 1 - 2.59T + 13T^{2} \)
17 \( 1 + 3.00T + 17T^{2} \)
19 \( 1 + 6.44T + 19T^{2} \)
23 \( 1 + 6.65T + 23T^{2} \)
29 \( 1 - 0.745T + 29T^{2} \)
31 \( 1 + 6.77T + 31T^{2} \)
37 \( 1 - 3.42T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 9.67T + 43T^{2} \)
47 \( 1 - 1.71T + 47T^{2} \)
53 \( 1 + 1.95T + 53T^{2} \)
59 \( 1 - 2.01T + 59T^{2} \)
61 \( 1 - 4.02T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 - 1.82T + 73T^{2} \)
79 \( 1 - 7.86T + 79T^{2} \)
83 \( 1 + 0.167T + 83T^{2} \)
89 \( 1 + 3.06T + 89T^{2} \)
97 \( 1 + 2.34T + 97T^{2} \)
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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

−7.904026774964655495704677889818, −6.81091206526556125410169582539, −6.39383885521769871022927679800, −5.64498413087394114540770245433, −5.23925295646691588332804957629, −4.45909493670335800464525654132, −3.93499665600209956892693196252, −2.27624362946955968923254478416, −1.69290546059315795829881972161, −0.51613898318832568717081311774, 0.51613898318832568717081311774, 1.69290546059315795829881972161, 2.27624362946955968923254478416, 3.93499665600209956892693196252, 4.45909493670335800464525654132, 5.23925295646691588332804957629, 5.64498413087394114540770245433, 6.39383885521769871022927679800, 6.81091206526556125410169582539, 7.904026774964655495704677889818

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