Properties

Label 2-3920-1.1-c1-0-80
Degree $2$
Conductor $3920$
Sign $-1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.87·3-s − 5-s + 5.29·9-s − 1.46·11-s − 2.22·13-s − 2.87·15-s − 7.26·17-s − 5.48·19-s − 2.51·23-s + 25-s + 6.60·27-s − 7.12·29-s − 6.51·31-s − 4.22·33-s + 6.90·37-s − 6.39·39-s + 11.3·41-s − 3.31·43-s − 5.29·45-s − 8.36·47-s − 20.9·51-s − 7.00·53-s + 1.46·55-s − 15.8·57-s + 9.07·59-s − 11.2·61-s + 2.22·65-s + ⋯
L(s)  = 1  + 1.66·3-s − 0.447·5-s + 1.76·9-s − 0.441·11-s − 0.616·13-s − 0.743·15-s − 1.76·17-s − 1.25·19-s − 0.524·23-s + 0.200·25-s + 1.27·27-s − 1.32·29-s − 1.17·31-s − 0.734·33-s + 1.13·37-s − 1.02·39-s + 1.76·41-s − 0.505·43-s − 0.789·45-s − 1.22·47-s − 2.93·51-s − 0.962·53-s + 0.197·55-s − 2.09·57-s + 1.18·59-s − 1.43·61-s + 0.275·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good3 \( 1 - 2.87T + 3T^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
13 \( 1 + 2.22T + 13T^{2} \)
17 \( 1 + 7.26T + 17T^{2} \)
19 \( 1 + 5.48T + 19T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 + 6.51T + 31T^{2} \)
37 \( 1 - 6.90T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 3.31T + 43T^{2} \)
47 \( 1 + 8.36T + 47T^{2} \)
53 \( 1 + 7.00T + 53T^{2} \)
59 \( 1 - 9.07T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 6.41T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + 9.83T + 89T^{2} \)
97 \( 1 - 2.09T + 97T^{2} \)
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

−8.099992612805886704907524597409, −7.60140026752400967561298085016, −6.91701937561758439111637411029, −6.00527767061376915983765883787, −4.69501658284931436752912629323, −4.17398054714658365197633070452, −3.38144591894135661928402628872, −2.37212688521159150142968489047, −1.97413424286168116531305493673, 0, 1.97413424286168116531305493673, 2.37212688521159150142968489047, 3.38144591894135661928402628872, 4.17398054714658365197633070452, 4.69501658284931436752912629323, 6.00527767061376915983765883787, 6.91701937561758439111637411029, 7.60140026752400967561298085016, 8.099992612805886704907524597409

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