Properties

Label 2-8020-1.1-c1-0-53
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  + 2.73·3-s − 5-s − 3.39·7-s + 4.49·9-s + 2.78·11-s + 4.32·13-s − 2.73·15-s − 1.50·17-s − 3.42·19-s − 9.28·21-s + 3.48·23-s + 25-s + 4.08·27-s + 9.62·29-s − 5.68·31-s + 7.61·33-s + 3.39·35-s − 1.92·37-s + 11.8·39-s + 6.98·41-s − 2.45·43-s − 4.49·45-s − 5.87·47-s + 4.50·49-s − 4.11·51-s + 5.66·53-s − 2.78·55-s + ⋯
L(s)  = 1  + 1.58·3-s − 0.447·5-s − 1.28·7-s + 1.49·9-s + 0.838·11-s + 1.19·13-s − 0.706·15-s − 0.364·17-s − 0.786·19-s − 2.02·21-s + 0.726·23-s + 0.200·25-s + 0.786·27-s + 1.78·29-s − 1.02·31-s + 1.32·33-s + 0.573·35-s − 0.316·37-s + 1.89·39-s + 1.09·41-s − 0.374·43-s − 0.669·45-s − 0.857·47-s + 0.643·49-s − 0.576·51-s + 0.777·53-s − 0.375·55-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

\(L(1)\) \(\approx\) \(3.274295315\)
\(L(\frac12)\) \(\approx\) \(3.274295315\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 3T^{2} \)
7 \( 1 + 3.39T + 7T^{2} \)
11 \( 1 - 2.78T + 11T^{2} \)
13 \( 1 - 4.32T + 13T^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + 3.42T + 19T^{2} \)
23 \( 1 - 3.48T + 23T^{2} \)
29 \( 1 - 9.62T + 29T^{2} \)
31 \( 1 + 5.68T + 31T^{2} \)
37 \( 1 + 1.92T + 37T^{2} \)
41 \( 1 - 6.98T + 41T^{2} \)
43 \( 1 + 2.45T + 43T^{2} \)
47 \( 1 + 5.87T + 47T^{2} \)
53 \( 1 - 5.66T + 53T^{2} \)
59 \( 1 + 5.44T + 59T^{2} \)
61 \( 1 - 7.60T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + 3.20T + 71T^{2} \)
73 \( 1 - 2.58T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 - 5.20T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−8.026127795687413003176300914419, −7.11480086682493041049423975817, −6.61866328784676669167833650337, −6.04910737257088363627566976612, −4.73654418104657615164855952225, −3.87790419654604447361229488113, −3.52412489256943359931320874516, −2.86807979862911048456588082131, −1.97515291917803273716833028935, −0.834558889788229773135878586604, 0.834558889788229773135878586604, 1.97515291917803273716833028935, 2.86807979862911048456588082131, 3.52412489256943359931320874516, 3.87790419654604447361229488113, 4.73654418104657615164855952225, 6.04910737257088363627566976612, 6.61866328784676669167833650337, 7.11480086682493041049423975817, 8.026127795687413003176300914419

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