↑

Properties

Label 2-84-28.3-c1-0-4
Degree $2$
Conductor $84$
Sign $0.638 - 0.769i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
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  + (0.553 + 1.30i)2-s + (0.5 − 0.866i)3-s + (−1.38 + 1.44i)4-s + (0.834 − 0.481i)5-s + (1.40 + 0.171i)6-s + (1.20 + 2.35i)7-s + (−2.64 − 1.00i)8-s + (−0.499 − 0.866i)9-s + (1.08 + 0.819i)10-s + (−4.74 − 2.74i)11-s + (0.553 + 1.92i)12-s − 3.75i·13-s + (−2.40 + 2.86i)14-s − 0.963i·15-s + (−0.147 − 3.99i)16-s + (−0.594 − 0.343i)17-s + ⋯
L(s)  = 1  + (0.391 + 0.920i)2-s + (0.288 − 0.499i)3-s + (−0.693 + 0.720i)4-s + (0.373 − 0.215i)5-s + (0.573 + 0.0700i)6-s + (0.453 + 0.891i)7-s + (−0.934 − 0.356i)8-s + (−0.166 − 0.288i)9-s + (0.344 + 0.259i)10-s + (−1.43 − 0.826i)11-s + (0.159 + 0.554i)12-s − 1.04i·13-s + (−0.642 + 0.766i)14-s − 0.248i·15-s + (−0.0369 − 0.999i)16-s + (−0.144 − 0.0832i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.638 - 0.769i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06975 + 0.502457i\)
\(L(\frac12)\) \(\approx\) \(1.06975 + 0.502457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.553 - 1.30i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.20 - 2.35i)T \)
good5 \( 1 + (-0.834 + 0.481i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.74 + 2.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.75iT - 13T^{2} \)
17 \( 1 + (0.594 + 0.343i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.44 - 4.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.07 + 0.620i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 + (-2.41 + 4.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.36 - 2.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.42iT - 41T^{2} \)
43 \( 1 - 5.97iT - 43T^{2} \)
47 \( 1 + (-1.80 - 3.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.04 + 3.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.34 + 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.01 + 5.20i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.17 + 4.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (5.76 + 3.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.22 - 0.707i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.543T + 83T^{2} \)
89 \( 1 + (-0.480 + 0.277i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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

−14.49054529717512781859990708269, −13.29651837168722452713463881168, −12.82938367091031545779349714448, −11.46781421319337776265924412790, −9.700345851384362094341267837965, −8.307230740322969498036362617479, −7.80964686447910145200975980040, −5.97324229688518232702205250684, −5.24578657898639484590978117717, −2.96801562536090375430883642416, 2.35976836196472391233673785529, 4.15933427129006083675258968441, 5.23466620581489199000572578765, 7.23889164569362100279602592175, 8.889818113627612119867174198497, 10.10259921838374615518847751409, 10.68148255099762367157945451889, 11.87073124676110188035354614545, 13.30148199079604510951986617509, 13.87706438420434866849942305097

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