Properties

Label 2-3330-5.4-c1-0-65
Degree $2$
Conductor $3330$
Sign $-0.894 + 0.447i$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
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  i·2-s − 4-s + (2 − i)5-s + 4.44i·7-s + i·8-s + (−1 − 2i)10-s − 4.89·11-s − 4i·13-s + 4.44·14-s + 16-s + 4.89i·17-s − 3.55·19-s + (−2 + i)20-s + 4.89i·22-s − 8.89i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.894 − 0.447i)5-s + 1.68i·7-s + 0.353i·8-s + (−0.316 − 0.632i)10-s − 1.47·11-s − 1.10i·13-s + 1.18·14-s + 0.250·16-s + 1.18i·17-s − 0.814·19-s + (−0.447 + 0.223i)20-s + 1.04i·22-s − 1.85i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9423651109\)
\(L(\frac12)\) \(\approx\) \(0.9423651109\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-2 + i)T \)
37 \( 1 - iT \)
good7 \( 1 - 4.44iT - 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 + 8.89iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 1.55T + 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 4.44iT - 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 5.55iT - 67T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 6.44T + 79T^{2} \)
83 \( 1 + 9.55iT - 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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

−8.470234932569464749564263207605, −8.070320595015816344953832180730, −6.52832866763828253755500551619, −5.77877858393814011771884109636, −5.31865263381309163936451456359, −4.61689160070126557307331085074, −3.22733608796866160349886412763, −2.37741885935425227188931316570, −1.99117567434106259796710959050, −0.27850336471684107560193235177, 1.25943692057206533591662828052, 2.49637824876114744802667501522, 3.56034688239445638859493665558, 4.49037499901271941204917541102, 5.19534768082190336096555025724, 6.01346689259205706839601122809, 6.91682246303763663191734926525, 7.31361260319650335360375324192, 7.87797915567083870422268963995, 9.014213980143646125218489777002

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