Properties

Label 2-8020-1.1-c1-0-27
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  + 0.732·3-s − 5-s + 3.46·7-s − 2.46·9-s − 3.46·11-s − 3.26·13-s − 0.732·15-s − 0.732·17-s − 6.92·19-s + 2.53·21-s − 2.19·23-s + 25-s − 4·27-s − 1.46·29-s + 1.46·31-s − 2.53·33-s − 3.46·35-s + 10.1·37-s − 2.39·39-s + 8.39·41-s − 2·43-s + 2.46·45-s + 4.92·47-s + 4.99·49-s − 0.535·51-s + 1.80·53-s + 3.46·55-s + ⋯
L(s)  = 1  + 0.422·3-s − 0.447·5-s + 1.30·7-s − 0.821·9-s − 1.04·11-s − 0.906·13-s − 0.189·15-s − 0.177·17-s − 1.58·19-s + 0.553·21-s − 0.457·23-s + 0.200·25-s − 0.769·27-s − 0.271·29-s + 0.262·31-s − 0.441·33-s − 0.585·35-s + 1.67·37-s − 0.383·39-s + 1.31·41-s − 0.304·43-s + 0.367·45-s + 0.718·47-s + 0.714·49-s − 0.0750·51-s + 0.247·53-s + 0.467·55-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

\(L(1)\) \(\approx\) \(1.566746697\)
\(L(\frac12)\) \(\approx\) \(1.566746697\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732T + 3T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 3.26T + 13T^{2} \)
17 \( 1 + 0.732T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + 2.19T + 23T^{2} \)
29 \( 1 + 1.46T + 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 8.39T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 - 4.92T + 47T^{2} \)
53 \( 1 - 1.80T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 - 2.19T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 7.26T + 97T^{2} \)
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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

−7.74733845697301178225989411630, −7.63079997405439544254797238383, −6.39004813766133549467414595153, −5.72236520779629232424433432274, −4.81313050260009395957652222512, −4.49391564219667765674273899465, −3.48958739474804248679747300250, −2.33157528550292942006398605068, −2.22968582082182138094575734697, −0.57271194669710679835503882252, 0.57271194669710679835503882252, 2.22968582082182138094575734697, 2.33157528550292942006398605068, 3.48958739474804248679747300250, 4.49391564219667765674273899465, 4.81313050260009395957652222512, 5.72236520779629232424433432274, 6.39004813766133549467414595153, 7.63079997405439544254797238383, 7.74733845697301178225989411630

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