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Properties

Label 2-864-36.23-c1-0-0
Degree $2$
Conductor $864$
Sign $-0.745 - 0.665i$
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  + (−3.40 − 1.96i)5-s + (0.961 − 0.555i)7-s + (−1.63 − 2.82i)11-s + (0.124 − 0.216i)13-s + 5.86i·17-s + 2.19i·19-s + (−2.79 + 4.83i)23-s + (5.24 + 9.09i)25-s + (−2.35 + 1.36i)29-s + (−8.96 − 5.17i)31-s − 4.36·35-s − 0.333·37-s + (5.28 + 3.05i)41-s + (−8.50 + 4.91i)43-s + (−4.70 − 8.15i)47-s + ⋯
L(s)  = 1  + (−1.52 − 0.880i)5-s + (0.363 − 0.209i)7-s + (−0.492 − 0.852i)11-s + (0.0346 − 0.0600i)13-s + 1.42i·17-s + 0.504i·19-s + (−0.582 + 1.00i)23-s + (1.04 + 1.81i)25-s + (−0.437 + 0.252i)29-s + (−1.61 − 0.930i)31-s − 0.738·35-s − 0.0547·37-s + (0.825 + 0.476i)41-s + (−1.29 + 0.749i)43-s + (−0.686 − 1.18i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.745 - 0.665i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.745 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0405548 + 0.106317i\)
\(L(\frac12)\) \(\approx\) \(0.0405548 + 0.106317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.40 + 1.96i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.961 + 0.555i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.63 + 2.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.124 + 0.216i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.86iT - 17T^{2} \)
19 \( 1 - 2.19iT - 19T^{2} \)
23 \( 1 + (2.79 - 4.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.35 - 1.36i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.96 + 5.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.333T + 37T^{2} \)
41 \( 1 + (-5.28 - 3.05i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.50 - 4.91i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.70 + 8.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.75iT - 53T^{2} \)
59 \( 1 + (-3.26 + 5.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.07 - 1.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.501 - 0.289i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.26T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + (-7.67 + 4.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.34 - 4.05i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.75iT - 89T^{2} \)
97 \( 1 + (-0.916 - 1.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.74067569402300253247232063781, −9.538175606486099924946054736288, −8.502841824499984230607154720889, −8.033292339529621458736723039475, −7.44648503260068061218908446281, −6.01117501915379082547528589055, −5.13280209591025177042352310721, −4.01982159682020344403934197216, −3.49174494290199956796230281745, −1.51972770955554038936915714467, 0.05554762654532439259967034847, 2.32022333733027716907239890756, 3.35656199859243176988147216783, 4.39599041420846997060040383866, 5.21740087513046454701645109104, 6.75268071149241807692206347526, 7.27929480669359799450127926212, 7.974012903208278658086861227436, 8.886822147966011986014092636526, 9.965054288065373628811821181117

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