↑

Properties

Label 2-855-95.64-c1-0-26
Degree $2$
Conductor $855$
Sign $0.936 - 0.350i$
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  + (−0.330 + 0.190i)2-s + (−0.927 + 1.60i)4-s + (1.93 − 1.11i)5-s − 1.47i·8-s + (−0.427 + 0.739i)10-s + (−1.57 − 2.72i)16-s + (3.25 − 1.88i)17-s + (4.35 − 0.204i)19-s + 4.14i·20-s + (7.74 + 4.47i)23-s + (2.5 − 4.33i)25-s − 10.7·31-s + (3.59 + 2.07i)32-s + (−0.718 + 1.24i)34-s + (−1.40 + 0.899i)38-s + ⋯
L(s)  = 1  + (−0.233 + 0.135i)2-s + (−0.463 + 0.802i)4-s + (0.866 − 0.499i)5-s − 0.520i·8-s + (−0.135 + 0.233i)10-s + (−0.393 − 0.681i)16-s + (0.790 − 0.456i)17-s + (0.998 − 0.0469i)19-s + 0.927i·20-s + (1.61 + 0.932i)23-s + (0.5 − 0.866i)25-s − 1.92·31-s + (0.634 + 0.366i)32-s + (−0.123 + 0.213i)34-s + (−0.227 + 0.145i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.936 - 0.350i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47037 + 0.265790i\)
\(L(\frac12)\) \(\approx\) \(1.47037 + 0.265790i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 + 1.11i)T \)
19 \( 1 + (-4.35 + 0.204i)T \)
good2 \( 1 + (0.330 - 0.190i)T + (1 - 1.73i)T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.25 + 1.88i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.74 - 4.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.92 - 2.26i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-12.3 - 7.11i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (48.5 - 84.0i)T^{2} \)
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

−9.924283091413820071429624507875, −9.220260019095715215585837029630, −8.814196866082995045547349089733, −7.57663520728058025446930124378, −7.10179838005469483419021395242, −5.63987738727269460561433876001, −5.06932362549599749896258821218, −3.79296739355685026482315176861, −2.76448345729145887154027552852, −1.10259100839080359323870186373, 1.10619225800493011053447769347, 2.33950377324641261189736313072, 3.62943335366472803880030975349, 5.12805555413258862380441945829, 5.58820395369272780292614376483, 6.61285329736913444616003230512, 7.50759646225152794878114407404, 8.831066203826035455481808992908, 9.288267339552234760723811309794, 10.23345339584839596483285280597

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