↑

Properties

Label 2-420-60.59-c1-0-27
Degree $2$
Conductor $420$
Sign $0.923 - 0.382i$
Analytic cond. $3.35371$
Root an. cond. $1.83131$
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  + (1.40 + 0.114i)2-s + (−1.47 − 0.909i)3-s + (1.97 + 0.321i)4-s + (−1 + 2i)5-s + (−1.97 − 1.45i)6-s + 7-s + (2.74 + 0.678i)8-s + (1.34 + 2.68i)9-s + (−1.63 + 2.70i)10-s + 2.94·11-s + (−2.61 − 2.26i)12-s − 1.36i·13-s + (1.40 + 0.114i)14-s + (3.29 − 2.03i)15-s + (3.79 + 1.26i)16-s + 2.69·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0806i)2-s + (−0.851 − 0.525i)3-s + (0.986 + 0.160i)4-s + (−0.447 + 0.894i)5-s + (−0.805 − 0.592i)6-s + 0.377·7-s + (0.970 + 0.239i)8-s + (0.448 + 0.893i)9-s + (−0.517 + 0.855i)10-s + 0.888·11-s + (−0.755 − 0.655i)12-s − 0.378i·13-s + (0.376 + 0.0304i)14-s + (0.850 − 0.526i)15-s + (0.948 + 0.317i)16-s + 0.652·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98568 + 0.395039i\)
\(L(\frac12)\) \(\approx\) \(1.98568 + 0.395039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.114i)T \)
3 \( 1 + (1.47 + 0.909i)T \)
5 \( 1 + (1 - 2i)T \)
7 \( 1 - T \)
good11 \( 1 - 2.94T + 11T^{2} \)
13 \( 1 + 1.36iT - 13T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 - 3.54iT - 19T^{2} \)
23 \( 1 - 7.18iT - 23T^{2} \)
29 \( 1 + 1.36iT - 29T^{2} \)
31 \( 1 + 8.09iT - 31T^{2} \)
37 \( 1 + 9.89iT - 37T^{2} \)
41 \( 1 - 5.89iT - 41T^{2} \)
43 \( 1 + 2.25T + 43T^{2} \)
47 \( 1 - 1.81iT - 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + 6.80T + 59T^{2} \)
61 \( 1 + 6.55T + 61T^{2} \)
67 \( 1 + 8.46T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 16.1iT - 79T^{2} \)
83 \( 1 + 5.83iT - 83T^{2} \)
89 \( 1 + 4.83iT - 89T^{2} \)
97 \( 1 - 3.70iT - 97T^{2} \)
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

−11.46712428722480576348020082235, −10.83767091235473972575092699220, −9.804379821663452975278508854767, −7.78072379522927369494681389874, −7.50616170608391908525868424293, −6.27463537853945331669333104269, −5.73838475125243064401978156799, −4.40969627050858980169701041735, −3.34467753566569263159818840667, −1.73184245860803363294287008278, 1.29347478524658942069749144785, 3.43617139215066148435000328783, 4.58402468887878953331724057701, 4.95985606998334334382238879891, 6.22049264875913366629448744071, 7.03496180867493682159260243221, 8.415839696474104870664779406083, 9.466834878463010535836639559077, 10.60604048115811034649339325121, 11.33831428372125796082973898720

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