Properties

Label 2-8020-1.1-c1-0-44
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  + 1.75·3-s − 5-s + 0.706·7-s + 0.0928·9-s − 0.467·11-s − 0.643·13-s − 1.75·15-s + 5.29·17-s + 4.59·19-s + 1.24·21-s − 8.36·23-s + 25-s − 5.11·27-s + 0.263·29-s + 2.56·31-s − 0.822·33-s − 0.706·35-s + 9.10·37-s − 1.13·39-s + 4.62·41-s − 3.61·43-s − 0.0928·45-s − 5.16·47-s − 6.50·49-s + 9.31·51-s + 11.0·53-s + 0.467·55-s + ⋯
L(s)  = 1  + 1.01·3-s − 0.447·5-s + 0.267·7-s + 0.0309·9-s − 0.141·11-s − 0.178·13-s − 0.454·15-s + 1.28·17-s + 1.05·19-s + 0.271·21-s − 1.74·23-s + 0.200·25-s − 0.983·27-s + 0.0489·29-s + 0.460·31-s − 0.143·33-s − 0.119·35-s + 1.49·37-s − 0.181·39-s + 0.722·41-s − 0.551·43-s − 0.0138·45-s − 0.753·47-s − 0.928·49-s + 1.30·51-s + 1.51·53-s + 0.0631·55-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

\(L(1)\) \(\approx\) \(2.722975172\)
\(L(\frac12)\) \(\approx\) \(2.722975172\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.75T + 3T^{2} \)
7 \( 1 - 0.706T + 7T^{2} \)
11 \( 1 + 0.467T + 11T^{2} \)
13 \( 1 + 0.643T + 13T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 4.59T + 19T^{2} \)
23 \( 1 + 8.36T + 23T^{2} \)
29 \( 1 - 0.263T + 29T^{2} \)
31 \( 1 - 2.56T + 31T^{2} \)
37 \( 1 - 9.10T + 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 + 3.61T + 43T^{2} \)
47 \( 1 + 5.16T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + 1.05T + 59T^{2} \)
61 \( 1 - 6.45T + 61T^{2} \)
67 \( 1 + 5.75T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 - 0.301T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 9.95T + 97T^{2} \)
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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

−7.85824628326983632802653866589, −7.58094952683296560555660346258, −6.48435964579294106627579965208, −5.71088510300547701019423815223, −5.01914500250426959162961803214, −4.05219302325013612832846412788, −3.46957670164297083208863511683, −2.75779832800197554079556465183, −1.94208527380319129303731593970, −0.77560042546034993827727201609, 0.77560042546034993827727201609, 1.94208527380319129303731593970, 2.75779832800197554079556465183, 3.46957670164297083208863511683, 4.05219302325013612832846412788, 5.01914500250426959162961803214, 5.71088510300547701019423815223, 6.48435964579294106627579965208, 7.58094952683296560555660346258, 7.85824628326983632802653866589

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