↑

Properties

Label 2-1680-105.104-c1-0-28
Degree $2$
Conductor $1680$
Sign $0.656 - 0.754i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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  + (−0.420 − 1.68i)3-s + (1.95 + 1.08i)5-s + (2.37 + 1.16i)7-s + (−2.64 + 1.41i)9-s + 2.82i·11-s − 0.841·13-s + (1 − 3.74i)15-s + 1.19i·17-s + 4.55i·19-s + (0.955 − 4.48i)21-s − 3.29·23-s + (2.64 + 4.24i)25-s + (3.48 + 3.85i)27-s + 7.98i·29-s + 5.53i·31-s + ⋯
L(s)  = 1  + (−0.242 − 0.970i)3-s + (0.874 + 0.485i)5-s + (0.898 + 0.439i)7-s + (−0.881 + 0.471i)9-s + 0.852i·11-s − 0.233·13-s + (0.258 − 0.966i)15-s + 0.288i·17-s + 1.04i·19-s + (0.208 − 0.978i)21-s − 0.686·23-s + (0.529 + 0.848i)25-s + (0.671 + 0.740i)27-s + 1.48i·29-s + 0.993i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.714677617\)
\(L(\frac12)\) \(\approx\) \(1.714677617\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.420 + 1.68i)T \)
5 \( 1 + (-1.95 - 1.08i)T \)
7 \( 1 + (-2.37 - 1.16i)T \)
good11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 0.841T + 13T^{2} \)
17 \( 1 - 1.19iT - 17T^{2} \)
19 \( 1 - 4.55iT - 19T^{2} \)
23 \( 1 + 3.29T + 23T^{2} \)
29 \( 1 - 7.98iT - 29T^{2} \)
31 \( 1 - 5.53iT - 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + 4.65iT - 43T^{2} \)
47 \( 1 - 4.33iT - 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + 3.91T + 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 - 4.65iT - 67T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 - 3.06T + 73T^{2} \)
79 \( 1 - 7.29T + 79T^{2} \)
83 \( 1 + 7.70iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 8.11T + 97T^{2} \)
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

−9.360622597512210489702557401101, −8.610766941753512551252393938692, −7.70995662456452257923681476535, −7.11611416044779860145799207768, −6.22607366257061050508430222807, −5.52985959844349311846783808043, −4.79103056301568967359117293406, −3.26006061203373405830430320400, −2.00478314240859070692869767723, −1.64151290993407015060138171463, 0.65799494180351189158279591860, 2.15797628052521381129449966223, 3.33718308213204057249264165099, 4.55856743352415819651303959453, 4.92929800989216015080669970420, 5.89076130835834411178506324255, 6.56866782068644889711514693411, 7.996180187356394641487231765756, 8.464382064444053434524848093720, 9.473343621714265529878721505877

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