Properties

Label 2-855-95.74-c1-0-41
Degree $2$
Conductor $855$
Sign $-0.405 + 0.914i$
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  + (1.20 + 0.212i)2-s + (−0.477 − 0.173i)4-s + (−2.13 + 0.670i)5-s + (3.28 − 1.89i)7-s + (−2.65 − 1.53i)8-s + (−2.70 + 0.354i)10-s + (−0.618 + 1.07i)11-s + (−2.22 − 2.64i)13-s + (4.35 − 1.58i)14-s + (−2.08 − 1.75i)16-s + (−2.96 − 0.522i)17-s + (−4.28 − 0.793i)19-s + (1.13 + 0.0503i)20-s + (−0.971 + 1.15i)22-s + (2.10 − 5.77i)23-s + ⋯
L(s)  = 1  + (0.850 + 0.149i)2-s + (−0.238 − 0.0868i)4-s + (−0.953 + 0.300i)5-s + (1.24 − 0.716i)7-s + (−0.937 − 0.541i)8-s + (−0.856 + 0.112i)10-s + (−0.186 + 0.323i)11-s + (−0.616 − 0.734i)13-s + (1.16 − 0.423i)14-s + (−0.522 − 0.438i)16-s + (−0.718 − 0.126i)17-s + (−0.983 − 0.181i)19-s + (0.253 + 0.0112i)20-s + (−0.207 + 0.246i)22-s + (0.438 − 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.405 + 0.914i$
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: \((2,\ 855,\ (\ :1/2),\ -0.405 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605786 - 0.931614i\)
\(L(\frac12)\) \(\approx\) \(0.605786 - 0.931614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 - 0.670i)T \)
19 \( 1 + (4.28 + 0.793i)T \)
good2 \( 1 + (-1.20 - 0.212i)T + (1.87 + 0.684i)T^{2} \)
7 \( 1 + (-3.28 + 1.89i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.618 - 1.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.22 + 2.64i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.96 + 0.522i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-2.10 + 5.77i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.744 + 4.22i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.55 + 4.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.13iT - 37T^{2} \)
41 \( 1 + (4.08 + 3.42i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.12 + 8.57i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-7.19 + 1.26i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.13 - 3.13i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.141 - 0.804i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (6.01 + 2.18i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.995 + 0.175i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-12.8 + 4.67i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (7.11 - 8.47i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-1.06 - 0.889i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.18 - 1.26i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.06 + 1.73i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-3.01 - 0.531i)T + (91.1 + 33.1i)T^{2} \)
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

−10.19537019560436373465926272362, −8.845908446899901294510750956561, −8.129487716180056163396303741138, −7.29142976677848410016605799935, −6.44656617192425844725733899736, −5.08845074097227650788140481096, −4.54351465768492678793117166434, −3.84693375570630325147158611481, −2.48428609994341642726631921018, −0.40325808669701484099962329664, 1.92795220766973573429303228591, 3.26704346099029811658951725215, 4.34058560710998127320136521832, 4.88956357194032307379129332491, 5.70179079345863738868613797657, 7.04649599737568898915888492231, 8.056468308449783112503727274259, 8.707731460630141540828997754302, 9.281894389917805198057877788019, 10.97282117976148350446362762216

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