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Properties

Label 2-864-72.61-c1-0-4
Degree $2$
Conductor $864$
Sign $0.00870 + 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.17 + 1.83i)5-s + (0.191 − 0.332i)7-s + (−1.73 − 1.00i)11-s + (−0.397 + 0.229i)13-s + 4.08·17-s − 4.72i·19-s + (−2.97 − 5.15i)23-s + (4.21 − 7.29i)25-s + (2.03 + 1.17i)29-s + (−0.592 − 1.02i)31-s + 1.40i·35-s − 5.74i·37-s + (−4.75 − 8.23i)41-s + (−1.03 − 0.598i)43-s + (3.27 − 5.67i)47-s + ⋯
L(s)  = 1  + (−1.41 + 0.819i)5-s + (0.0725 − 0.125i)7-s + (−0.524 − 0.302i)11-s + (−0.110 + 0.0636i)13-s + 0.990·17-s − 1.08i·19-s + (−0.620 − 1.07i)23-s + (0.842 − 1.45i)25-s + (0.378 + 0.218i)29-s + (−0.106 − 0.184i)31-s + 0.237i·35-s − 0.944i·37-s + (−0.742 − 1.28i)41-s + (−0.158 − 0.0912i)43-s + (0.477 − 0.827i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00870 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00870 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.00870 + 0.999i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.00870 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489992 - 0.485745i\)
\(L(\frac12)\) \(\approx\) \(0.489992 - 0.485745i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.17 - 1.83i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.191 + 0.332i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.73 + 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.397 - 0.229i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.08T + 17T^{2} \)
19 \( 1 + 4.72iT - 19T^{2} \)
23 \( 1 + (2.97 + 5.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.03 - 1.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.592 + 1.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.74iT - 37T^{2} \)
41 \( 1 + (4.75 + 8.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.03 + 0.598i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.27 + 5.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.63iT - 53T^{2} \)
59 \( 1 + (-0.603 + 0.348i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.23 - 2.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.87 - 5.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 + 2.68T + 73T^{2} \)
79 \( 1 + (-5.35 + 9.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.49 + 3.16i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.56T + 89T^{2} \)
97 \( 1 + (2.98 - 5.17i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.31273975616100355249331641496, −8.941476278229226460694397579771, −8.136774663401715841035898688719, −7.40055922295906535164518883813, −6.77008717740564744053646393239, −5.53213013915740105516893749102, −4.38812803880590777795082493603, −3.51125372402906481154294031234, −2.55259561951100054328194493106, −0.36098764616169816428228321360, 1.34851619606696013216876327493, 3.14708610506656714491453249391, 4.07368214123033751414011574021, 4.96740928490165404226572391258, 5.86854883614402531643607518452, 7.26589283496340398979802198890, 7.970543511551067272363921372457, 8.379407044378923179365006494420, 9.572436164929899524787514146754, 10.29325593767256662838918596336

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