↑

Properties

Label 2-3920-140.79-c0-0-3
Degree $2$
Conductor $3920$
Sign $0.968 - 0.250i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)9-s + (−0.499 − 0.866i)25-s + 2·29-s + 2·41-s + (0.499 + 0.866i)45-s + (−1 + 1.73i)61-s + (−0.499 − 0.866i)81-s + (1 − 1.73i)89-s + (1 + 1.73i)101-s + (1 + 1.73i)109-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)9-s + (−0.499 − 0.866i)25-s + 2·29-s + 2·41-s + (0.499 + 0.866i)45-s + (−1 + 1.73i)61-s + (−0.499 − 0.866i)81-s + (1 − 1.73i)89-s + (1 + 1.73i)101-s + (1 + 1.73i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.968 - 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243046349\)
\(L(\frac12)\) \(\approx\) \(1.243046349\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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

−8.702492116265349537293042649560, −7.77915737582202487182340114251, −7.24802992507568953803685822014, −6.44553819269419844744212034115, −6.00618644976148158386440807334, −4.71309788299406934760416163815, −4.06327471098678505270898339461, −3.21928571733310263959036936247, −2.45327454368982615807571498187, −0.990578571973092936812892467527, 0.976050005442044870313045094085, 2.08894897484033772793513713653, 3.17364617860842158216506235416, 4.31122266114823230233121323007, 4.68396227160948575842711348784, 5.51556473355296007245095231125, 6.42174566478572999440638066040, 7.32983875188193982980792942525, 7.955275465023248645659869494878, 8.476584581521341309333890046944

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