Properties

Label 2-8020-1.1-c1-0-35
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.35·3-s + 5-s − 1.59·7-s + 2.55·9-s − 2.22·11-s + 3.94·13-s − 2.35·15-s + 6.26·17-s + 3.37·19-s + 3.74·21-s + 4.60·23-s + 25-s + 1.04·27-s + 7.90·29-s + 1.30·31-s + 5.23·33-s − 1.59·35-s + 1.20·37-s − 9.28·39-s + 9.85·41-s − 4.13·43-s + 2.55·45-s − 3.91·47-s − 4.47·49-s − 14.7·51-s − 11.1·53-s − 2.22·55-s + ⋯
L(s)  = 1  − 1.36·3-s + 0.447·5-s − 0.601·7-s + 0.851·9-s − 0.669·11-s + 1.09·13-s − 0.608·15-s + 1.52·17-s + 0.774·19-s + 0.817·21-s + 0.960·23-s + 0.200·25-s + 0.201·27-s + 1.46·29-s + 0.234·31-s + 0.911·33-s − 0.268·35-s + 0.198·37-s − 1.48·39-s + 1.53·41-s − 0.630·43-s + 0.380·45-s − 0.570·47-s − 0.638·49-s − 2.06·51-s − 1.53·53-s − 0.299·55-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

\(L(1)\) \(\approx\) \(1.398827244\)
\(L(\frac12)\) \(\approx\) \(1.398827244\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.35T + 3T^{2} \)
7 \( 1 + 1.59T + 7T^{2} \)
11 \( 1 + 2.22T + 11T^{2} \)
13 \( 1 - 3.94T + 13T^{2} \)
17 \( 1 - 6.26T + 17T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
23 \( 1 - 4.60T + 23T^{2} \)
29 \( 1 - 7.90T + 29T^{2} \)
31 \( 1 - 1.30T + 31T^{2} \)
37 \( 1 - 1.20T + 37T^{2} \)
41 \( 1 - 9.85T + 41T^{2} \)
43 \( 1 + 4.13T + 43T^{2} \)
47 \( 1 + 3.91T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + 5.78T + 59T^{2} \)
61 \( 1 - 7.73T + 61T^{2} \)
67 \( 1 + 2.39T + 67T^{2} \)
71 \( 1 + 6.69T + 71T^{2} \)
73 \( 1 - 1.51T + 73T^{2} \)
79 \( 1 - 1.73T + 79T^{2} \)
83 \( 1 + 0.0337T + 83T^{2} \)
89 \( 1 + 5.17T + 89T^{2} \)
97 \( 1 - 8.62T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.74950910767488627946024218349, −6.92196041851485989790371874335, −6.22092921321723750447411415401, −5.87268278358449273549801157794, −5.16095601241092459582105003541, −4.61356578572279040090535009203, −3.35552404347692130398680487312, −2.88346559059303987264151407047, −1.36549935584738940296749745331, −0.70823955599326522044180717507, 0.70823955599326522044180717507, 1.36549935584738940296749745331, 2.88346559059303987264151407047, 3.35552404347692130398680487312, 4.61356578572279040090535009203, 5.16095601241092459582105003541, 5.87268278358449273549801157794, 6.22092921321723750447411415401, 6.92196041851485989790371874335, 7.74950910767488627946024218349

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