↑

Properties

Label 2-864-216.133-c1-0-19
Degree $2$
Conductor $864$
Sign $0.974 + 0.222i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
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  + (1.40 + 1.01i)3-s + (−1.55 − 1.85i)5-s + (−0.763 + 0.278i)7-s + (0.935 + 2.85i)9-s + (2.23 − 2.65i)11-s + (2.83 − 0.500i)13-s + (−0.299 − 4.18i)15-s + (−3.55 − 6.15i)17-s + (5.16 + 2.98i)19-s + (−1.35 − 0.386i)21-s + (6.58 + 2.39i)23-s + (−0.151 + 0.860i)25-s + (−1.58 + 4.94i)27-s + (6.59 + 1.16i)29-s + (−2.55 − 0.930i)31-s + ⋯
L(s)  = 1  + (0.809 + 0.586i)3-s + (−0.696 − 0.830i)5-s + (−0.288 + 0.105i)7-s + (0.311 + 0.950i)9-s + (0.672 − 0.801i)11-s + (0.787 − 0.138i)13-s + (−0.0772 − 1.08i)15-s + (−0.861 − 1.49i)17-s + (1.18 + 0.683i)19-s + (−0.295 − 0.0842i)21-s + (1.37 + 0.500i)23-s + (−0.0303 + 0.172i)25-s + (−0.304 + 0.952i)27-s + (1.22 + 0.215i)29-s + (−0.459 − 0.167i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.974 + 0.222i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.974 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90245 - 0.214729i\)
\(L(\frac12)\) \(\approx\) \(1.90245 - 0.214729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.40 - 1.01i)T \)
good5 \( 1 + (1.55 + 1.85i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.763 - 0.278i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.23 + 2.65i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-2.83 + 0.500i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.55 + 6.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.16 - 2.98i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.58 - 2.39i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-6.59 - 1.16i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.55 + 0.930i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-3.59 + 2.07i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.742 + 4.20i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.77 + 2.10i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (2.80 - 1.01i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 7.38iT - 53T^{2} \)
59 \( 1 + (-4.72 - 5.62i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.0900 - 0.247i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.20 - 0.388i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.69 + 4.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.97 - 3.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.76 - 10.0i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (7.86 + 1.38i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (-2.14 + 3.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.28 - 3.59i)T + (16.8 + 95.5i)T^{2} \)
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

−9.887472243038670135357609102062, −9.051932274350647734647864649142, −8.717124517025003632022422581773, −7.79810654188387090747385276269, −6.87021697133993784888362989009, −5.51100480140859259216625240695, −4.64250541944110806662044771248, −3.68505963727119365836787547675, −2.89760794569798109886000306085, −1.03656852331068779717156209792, 1.36217850636733004547900535321, 2.80163244218056463661216533656, 3.60160735715702710462613117509, 4.54041491128289960996514067773, 6.35166728860567100086708298725, 6.80011948062830841112364773794, 7.55664010706852592980972679534, 8.507095163229698969389820633550, 9.179306133708754103287968450010, 10.14775348472312353783138533839

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