↑

Properties

Label 2-864-9.4-c1-0-9
Degree $2$
Conductor $864$
Sign $0.173 + 0.984i$
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  + (2 − 3.46i)5-s + (−1 − 1.73i)7-s + (2.5 + 4.33i)11-s + (1 − 1.73i)13-s + 3·17-s − 19-s + (3 − 5.19i)23-s + (−5.49 − 9.52i)25-s + (−1 − 1.73i)29-s + (−2 + 3.46i)31-s − 7.99·35-s − 8·37-s + (0.5 − 0.866i)41-s + (−3.5 − 6.06i)43-s + (−1 − 1.73i)47-s + ⋯
L(s)  = 1  + (0.894 − 1.54i)5-s + (−0.377 − 0.654i)7-s + (0.753 + 1.30i)11-s + (0.277 − 0.480i)13-s + 0.727·17-s − 0.229·19-s + (0.625 − 1.08i)23-s + (−1.09 − 1.90i)25-s + (−0.185 − 0.321i)29-s + (−0.359 + 0.622i)31-s − 1.35·35-s − 1.31·37-s + (0.0780 − 0.135i)41-s + (−0.533 − 0.924i)43-s + (−0.145 − 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36985 - 1.14944i\)
\(L(\frac12)\) \(\approx\) \(1.36985 - 1.14944i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.04182656772587724229844475071, −9.076752826957541386120193771066, −8.578359924860229690609078581517, −7.36415557397249395309778182779, −6.50864501149247461424921089327, −5.42858096037832317312738873693, −4.71111885715002051973129873200, −3.74027243782593038455786808861, −2.01426221908549475756113576430, −0.931050050523385830925994661487, 1.72938968032600947888624752631, 3.03030657342701107608197841606, 3.56462638989247576135110153244, 5.41742110606744453734694803234, 6.13892645052954953804508974511, 6.66102425193647582214758345342, 7.68377632115047883804001126825, 8.945848211396252163943350247444, 9.465739342150552239219420179480, 10.38016479341323186450021173815

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