Properties

Label 2-864-288.11-c1-0-44
Degree $2$
Conductor $864$
Sign $-0.887 + 0.459i$
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  + (1.13 − 0.841i)2-s + (0.583 − 1.91i)4-s + (−0.335 − 2.55i)5-s + (−0.254 − 0.949i)7-s + (−0.947 − 2.66i)8-s + (−2.52 − 2.61i)10-s + (0.837 − 0.642i)11-s + (−1.15 + 1.50i)13-s + (−1.08 − 0.864i)14-s + (−3.31 − 2.23i)16-s − 4.36·17-s + (0.0481 − 0.0199i)19-s + (−5.07 − 0.845i)20-s + (0.410 − 1.43i)22-s + (−0.476 − 0.127i)23-s + ⋯
L(s)  = 1  + (0.803 − 0.595i)2-s + (0.291 − 0.956i)4-s + (−0.150 − 1.14i)5-s + (−0.0961 − 0.358i)7-s + (−0.334 − 0.942i)8-s + (−0.799 − 0.827i)10-s + (0.252 − 0.193i)11-s + (−0.320 + 0.418i)13-s + (−0.290 − 0.231i)14-s + (−0.829 − 0.557i)16-s − 1.05·17-s + (0.0110 − 0.00457i)19-s + (−1.13 − 0.188i)20-s + (0.0876 − 0.305i)22-s + (−0.0994 − 0.0266i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.887 + 0.459i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.887 + 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486857 - 1.99831i\)
\(L(\frac12)\) \(\approx\) \(0.486857 - 1.99831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 0.841i)T \)
3 \( 1 \)
good5 \( 1 + (0.335 + 2.55i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (0.254 + 0.949i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.837 + 0.642i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (1.15 - 1.50i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 + 4.36T + 17T^{2} \)
19 \( 1 + (-0.0481 + 0.0199i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.476 + 0.127i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-8.61 - 1.13i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (-4.82 + 2.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.37 - 8.14i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.939 + 3.50i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.85 + 8.92i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (2.36 + 1.36i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.72 - 6.57i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.89 - 0.776i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (-0.240 + 1.83i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-9.22 + 12.0i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-10.6 - 10.6i)T + 71iT^{2} \)
73 \( 1 + (-6.06 + 6.06i)T - 73iT^{2} \)
79 \( 1 + (-7.26 + 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.23 + 0.952i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (-3.67 + 3.67i)T - 89iT^{2} \)
97 \( 1 + (-2.21 + 3.82i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.957260172558550358062229016938, −9.035765310298872103436202462602, −8.342444387821528106808029920251, −6.94862646144178352160558991575, −6.23986630772640019574579625517, −4.92554534706843186462478511811, −4.56551906456716557837849906440, −3.46003673133224465130054546553, −2.08335566043395590689844204017, −0.75746752049685134087545498672, 2.40781972628515063812154737242, 3.16323089995316458906379139519, 4.30475692721350391928582188284, 5.25079549879371557509589241600, 6.52517003266736952403568471584, 6.68674719210094640754178079518, 7.81157138585947592350899153329, 8.567081625990706112367971845341, 9.702508393440503237529495332169, 10.72749476591656864707269685045

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