↑

Properties

Label 2-3330-185.184-c1-0-83
Degree $2$
Conductor $3330$
Sign $0.289 + 0.957i$
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  + 2-s + 4-s + (0.718 − 2.11i)5-s − 2.74i·7-s + 8-s + (0.718 − 2.11i)10-s + 0.572·11-s + 3.04·13-s − 2.74i·14-s + 16-s + 4.95·17-s + 3.74i·19-s + (0.718 − 2.11i)20-s + 0.572·22-s + 2.98·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.321 − 0.946i)5-s − 1.03i·7-s + 0.353·8-s + (0.227 − 0.669i)10-s + 0.172·11-s + 0.843·13-s − 0.734i·14-s + 0.250·16-s + 1.20·17-s + 0.859i·19-s + (0.160 − 0.473i)20-s + 0.122·22-s + 0.621·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 0.289 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.490698550\)
\(L(\frac12)\) \(\approx\) \(3.490698550\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + (-0.718 + 2.11i)T \)
37 \( 1 + (4.94 - 3.53i)T \)
good7 \( 1 + 2.74iT - 7T^{2} \)
11 \( 1 - 0.572T + 11T^{2} \)
13 \( 1 - 3.04T + 13T^{2} \)
17 \( 1 - 4.95T + 17T^{2} \)
19 \( 1 - 3.74iT - 19T^{2} \)
23 \( 1 - 2.98T + 23T^{2} \)
29 \( 1 + 7.36iT - 29T^{2} \)
31 \( 1 - 1.12iT - 31T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 6.91T + 43T^{2} \)
47 \( 1 - 0.776iT - 47T^{2} \)
53 \( 1 - 3.55iT - 53T^{2} \)
59 \( 1 - 9.27iT - 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 + 9.09iT - 67T^{2} \)
71 \( 1 + 6.26T + 71T^{2} \)
73 \( 1 - 7.83iT - 73T^{2} \)
79 \( 1 + 0.716iT - 79T^{2} \)
83 \( 1 + 2.01iT - 83T^{2} \)
89 \( 1 + 8.33iT - 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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

−8.264859269833452303237608298946, −7.79505286448097511747368039942, −6.89070892309519478572400895144, −5.99306611675195852211481026316, −5.49762924202322667823512048436, −4.48602745611107249496946778235, −3.95474764196877050545223900457, −3.09339134936924640505409214511, −1.66070410974902824224260270431, −0.907453490577889856936793002738, 1.42378023479272144170398235949, 2.55222070844939916695117796997, 3.15438840133752145800687142508, 3.96665349263200238679574964801, 5.27034767839467550612296109756, 5.60654111987243308876808260225, 6.48899777834118945841817242780, 7.04313529455665697439542784816, 7.910044111171461922715788720045, 8.866027324939333866094000071603

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