↑

Properties

Label 2-859-859.100-c1-0-41
Degree $2$
Conductor $859$
Sign $0.940 + 0.338i$
Analytic cond. $6.85914$
Root an. cond. $2.61899$
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  + (−2.04 + 1.07i)2-s + (−1.39 + 2.02i)3-s + (1.88 − 2.72i)4-s + (−0.257 − 0.372i)5-s + (0.684 − 5.63i)6-s + (0.629 + 5.18i)7-s + (−0.365 + 3.01i)8-s + (−1.08 − 2.85i)9-s + (0.923 + 0.484i)10-s + (−0.439 − 3.61i)11-s + (2.89 + 7.62i)12-s + 3.59·13-s + (−6.83 − 9.90i)14-s + 1.11·15-s + (−0.127 − 0.336i)16-s + (−2.95 + 0.727i)17-s + ⋯
L(s)  = 1  + (−1.44 + 0.757i)2-s + (−0.807 + 1.16i)3-s + (0.941 − 1.36i)4-s + (−0.114 − 0.166i)5-s + (0.279 − 2.29i)6-s + (0.237 + 1.95i)7-s + (−0.129 + 1.06i)8-s + (−0.361 − 0.953i)9-s + (0.292 + 0.153i)10-s + (−0.132 − 1.09i)11-s + (0.835 + 2.20i)12-s + 0.996·13-s + (−1.82 − 2.64i)14-s + 0.287·15-s + (−0.0319 − 0.0842i)16-s + (−0.716 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(859\)
Sign: $0.940 + 0.338i$
Analytic conductor: \(6.85914\)
Root analytic conductor: \(2.61899\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{859} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 859,\ (\ :1/2),\ 0.940 + 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123103 - 0.0214920i\)
\(L(\frac12)\) \(\approx\) \(0.123103 - 0.0214920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad859 \( 1 + (19.8 - 21.5i)T \)
good2 \( 1 + (2.04 - 1.07i)T + (1.13 - 1.64i)T^{2} \)
3 \( 1 + (1.39 - 2.02i)T + (-1.06 - 2.80i)T^{2} \)
5 \( 1 + (0.257 + 0.372i)T + (-1.77 + 4.67i)T^{2} \)
7 \( 1 + (-0.629 - 5.18i)T + (-6.79 + 1.67i)T^{2} \)
11 \( 1 + (0.439 + 3.61i)T + (-10.6 + 2.63i)T^{2} \)
13 \( 1 - 3.59T + 13T^{2} \)
17 \( 1 + (2.95 - 0.727i)T + (15.0 - 7.90i)T^{2} \)
19 \( 1 + 4.52T + 19T^{2} \)
23 \( 1 + (1.72 + 0.906i)T + (13.0 + 18.9i)T^{2} \)
29 \( 1 + (-1.17 + 3.09i)T + (-21.7 - 19.2i)T^{2} \)
31 \( 1 + (5.93 + 5.25i)T + (3.73 + 30.7i)T^{2} \)
37 \( 1 + (-0.341 - 0.900i)T + (-27.6 + 24.5i)T^{2} \)
41 \( 1 + (-5.63 + 8.15i)T + (-14.5 - 38.3i)T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + (-3.35 + 2.97i)T + (5.66 - 46.6i)T^{2} \)
53 \( 1 + (3.94 + 0.971i)T + (46.9 + 24.6i)T^{2} \)
59 \( 1 + (9.12 - 4.78i)T + (33.5 - 48.5i)T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + (-0.397 - 1.04i)T + (-50.1 + 44.4i)T^{2} \)
71 \( 1 + (-3.10 - 0.765i)T + (62.8 + 32.9i)T^{2} \)
73 \( 1 + (-0.257 + 2.11i)T + (-70.8 - 17.4i)T^{2} \)
79 \( 1 + (-0.592 + 0.857i)T + (-28.0 - 73.8i)T^{2} \)
83 \( 1 + (-0.166 - 1.37i)T + (-80.5 + 19.8i)T^{2} \)
89 \( 1 + (-10.6 - 2.62i)T + (78.8 + 41.3i)T^{2} \)
97 \( 1 + (1.62 - 0.401i)T + (85.8 - 45.0i)T^{2} \)
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

−9.959548973445716705668876633087, −9.002169903244772274467658427171, −8.689659195323293749442212489004, −8.063576781227712486940751344859, −6.30693096811703828562170248399, −6.04268187136094774435538797272, −5.20666712416836620513391167491, −3.92486004996365042556552463373, −2.18087009932759172739380968431, −0.12247919037494129218623435379, 1.17001310318864350040528899777, 1.82718081155612781256984438684, 3.55982472382333993399668896868, 4.79746173497369311258122315020, 6.49183044533990682543847892037, 7.07578257353856765048281259112, 7.60384568065884803510667877333, 8.433970251892898658844621323635, 9.567389895248767091969769019191, 10.43463059151578568117529293949

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