↑

Properties

Label 2-855-95.74-c1-0-17
Degree $2$
Conductor $855$
Sign $0.830 + 0.557i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
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  + (−1.45 − 0.256i)2-s + (0.176 + 0.0642i)4-s + (−0.658 − 2.13i)5-s + (2.81 − 1.62i)7-s + (2.32 + 1.34i)8-s + (0.410 + 3.28i)10-s + (−2.09 + 3.62i)11-s + (1.14 + 1.36i)13-s + (−4.51 + 1.64i)14-s + (−3.32 − 2.79i)16-s + (6.23 + 1.09i)17-s + (4.09 + 1.49i)19-s + (0.0210 − 0.419i)20-s + (3.97 − 4.74i)22-s + (−0.490 + 1.34i)23-s + ⋯
L(s)  = 1  + (−1.03 − 0.181i)2-s + (0.0883 + 0.0321i)4-s + (−0.294 − 0.955i)5-s + (1.06 − 0.614i)7-s + (0.820 + 0.473i)8-s + (0.129 + 1.03i)10-s + (−0.630 + 1.09i)11-s + (0.318 + 0.379i)13-s + (−1.20 + 0.439i)14-s + (−0.831 − 0.697i)16-s + (1.51 + 0.266i)17-s + (0.939 + 0.343i)19-s + (0.00470 − 0.0938i)20-s + (0.848 − 1.01i)22-s + (−0.102 + 0.280i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.830 + 0.557i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.830 + 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881926 - 0.268406i\)
\(L(\frac12)\) \(\approx\) \(0.881926 - 0.268406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.658 + 2.13i)T \)
19 \( 1 + (-4.09 - 1.49i)T \)
good2 \( 1 + (1.45 + 0.256i)T + (1.87 + 0.684i)T^{2} \)
7 \( 1 + (-2.81 + 1.62i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.09 - 3.62i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.14 - 1.36i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-6.23 - 1.09i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.490 - 1.34i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0589 + 0.334i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.38 - 2.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.70iT - 37T^{2} \)
41 \( 1 + (-5.46 - 4.58i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.38 + 9.29i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.438 + 0.0773i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-2.47 + 6.80i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.545 + 3.09i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-2.88 - 1.04i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-8.42 + 1.48i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-12.1 + 4.40i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-1.17 + 1.39i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.535 + 0.449i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.478 + 0.276i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.23 + 4.38i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-13.0 - 2.29i)T + (91.1 + 33.1i)T^{2} \)
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

−9.932359822875042011506633665225, −9.384533387714377864058903037548, −8.193488962262075278940437465613, −7.965013220794161300860900472866, −7.17986049043260343944144486025, −5.35911821121862731301569140591, −4.86386460101598070246738049489, −3.83325501051574259776729772068, −1.82201657734478215969666431230, −0.998438160662806111425971047943, 0.943397228757835315642196685309, 2.62869536412189338479129982460, 3.70165908200972888700492673516, 5.11870914614188084446714725219, 5.94874358549494928429439941141, 7.25142032724198742888087517313, 7.960907292146435418988912987038, 8.306697883335096630902997823367, 9.391567504446928138553298675080, 10.19439437009603021522047102739

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