Properties

Label 2-4020-201.200-c1-0-82
Degree $2$
Conductor $4020$
Sign $-0.943 + 0.330i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 + 1.33i)3-s + 5-s − 2.36i·7-s + (−0.540 − 2.95i)9-s − 0.927·11-s − 1.43i·13-s + (−1.10 + 1.33i)15-s − 5.72i·17-s − 1.85·19-s + (3.14 + 2.62i)21-s + 7.68i·23-s + 25-s + (4.52 + 2.55i)27-s − 7.94i·29-s + 0.732i·31-s + ⋯
L(s)  = 1  + (−0.640 + 0.768i)3-s + 0.447·5-s − 0.893i·7-s + (−0.180 − 0.983i)9-s − 0.279·11-s − 0.397i·13-s + (−0.286 + 0.343i)15-s − 1.38i·17-s − 0.426·19-s + (0.686 + 0.572i)21-s + 1.60i·23-s + 0.200·25-s + (0.870 + 0.491i)27-s − 1.47i·29-s + 0.131i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2483940194\)
\(L(\frac12)\) \(\approx\) \(0.2483940194\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.10 - 1.33i)T \)
5 \( 1 - T \)
67 \( 1 + (7.02 + 4.19i)T \)
good7 \( 1 + 2.36iT - 7T^{2} \)
11 \( 1 + 0.927T + 11T^{2} \)
13 \( 1 + 1.43iT - 13T^{2} \)
17 \( 1 + 5.72iT - 17T^{2} \)
19 \( 1 + 1.85T + 19T^{2} \)
23 \( 1 - 7.68iT - 23T^{2} \)
29 \( 1 + 7.94iT - 29T^{2} \)
31 \( 1 - 0.732iT - 31T^{2} \)
37 \( 1 + 1.54T + 37T^{2} \)
41 \( 1 + 6.32T + 41T^{2} \)
43 \( 1 - 5.59iT - 43T^{2} \)
47 \( 1 + 9.43iT - 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 6.47iT - 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
71 \( 1 + 5.24iT - 71T^{2} \)
73 \( 1 + 3.58T + 73T^{2} \)
79 \( 1 - 12.6iT - 79T^{2} \)
83 \( 1 - 4.64iT - 83T^{2} \)
89 \( 1 + 1.75iT - 89T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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

−8.057538982467282579219559418658, −7.30414024132988919937228891022, −6.60874058598281741270070101837, −5.73200793113645704542200671975, −5.15564811153461534537332166833, −4.38808625396567557845691431255, −3.58572131280934609392944548926, −2.69684346129017836521781187509, −1.24652186911167992448947421168, −0.07870970497146588275280039362, 1.53541426796723910309075196842, 2.15926575140209155459827897015, 3.14316018344458816199732637062, 4.48774533181444180278185075410, 5.16958953391251338175489112749, 6.01003203879456885175307668419, 6.40696231220909019682835094892, 7.13051966348892125946583652396, 8.203481578371148931515338615714, 8.556286632457795578572394859114

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