↑

Properties

Label 2-864-9.4-c1-0-11
Degree $2$
Conductor $864$
Sign $-0.851 + 0.524i$
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  + (1.18 − 2.05i)5-s + (−1.10 − 1.91i)7-s + (−2.96 − 5.14i)11-s + (−2.18 + 3.78i)13-s − 3.37·17-s − 3.72·19-s + (1.10 − 1.91i)23-s + (−0.313 − 0.543i)25-s + (0.186 + 0.322i)29-s + (−4.83 + 8.36i)31-s − 5.24·35-s + 4·37-s + (0.5 − 0.866i)41-s + (−2.96 − 5.14i)43-s + (1.10 + 1.91i)47-s + ⋯
L(s)  = 1  + (0.530 − 0.918i)5-s + (−0.417 − 0.723i)7-s + (−0.894 − 1.54i)11-s + (−0.606 + 1.05i)13-s − 0.817·17-s − 0.854·19-s + (0.230 − 0.399i)23-s + (−0.0627 − 0.108i)25-s + (0.0345 + 0.0598i)29-s + (−0.867 + 1.50i)31-s − 0.886·35-s + 0.657·37-s + (0.0780 − 0.135i)41-s + (−0.452 − 0.783i)43-s + (0.161 + 0.279i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.851 + 0.524i$
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: \((2,\ 864,\ (\ :1/2),\ -0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.240108 - 0.848223i\)
\(L(\frac12)\) \(\approx\) \(0.240108 - 0.848223i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 2.05i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.10 + 1.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.96 + 5.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.18 - 3.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.37T + 17T^{2} \)
19 \( 1 + 3.72T + 19T^{2} \)
23 \( 1 + (-1.10 + 1.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.186 - 0.322i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.83 - 8.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.96 + 5.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.10 - 1.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (-5.17 + 8.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.55 + 13.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.17 + 8.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 - 4.62T + 73T^{2} \)
79 \( 1 + (-4.83 - 8.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.04 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.25T + 89T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.756664386383716273293663496462, −8.872856453165756082833400820211, −8.411658342317378771423392017442, −7.14665282991601676502429763342, −6.35998661294261095643652031426, −5.32476392770231010450012743769, −4.52629487645441516705761067048, −3.34841834740955466904161535133, −1.97941630900436651521888111231, −0.39008996652937939048290724430, 2.32061620024358513698988837370, 2.65882681720687509840430133004, 4.27039395382880447455248488666, 5.36575586029675879954466866412, 6.17023849128379178779028666550, 7.12178368027145906834438658770, 7.79106342836815659821473464625, 8.984062960078319252517486405742, 9.879817724128057058327037350175, 10.30707812311313809924116317896

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