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Properties

Label 2-864-288.131-c1-0-17
Degree $2$
Conductor $864$
Sign $0.00109 - 0.999i$
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  + (1.40 + 0.191i)2-s + (1.92 + 0.537i)4-s + (−0.234 + 1.78i)5-s + (−0.927 + 3.45i)7-s + (2.59 + 1.12i)8-s + (−0.670 + 2.45i)10-s + (−0.296 − 0.227i)11-s + (−1.82 − 2.37i)13-s + (−1.96 + 4.66i)14-s + (3.42 + 2.07i)16-s − 1.74·17-s + (5.20 + 2.15i)19-s + (−1.40 + 3.30i)20-s + (−0.371 − 0.375i)22-s + (−6.63 + 1.77i)23-s + ⋯
L(s)  = 1  + (0.990 + 0.135i)2-s + (0.963 + 0.268i)4-s + (−0.104 + 0.797i)5-s + (−0.350 + 1.30i)7-s + (0.917 + 0.396i)8-s + (−0.212 + 0.775i)10-s + (−0.0892 − 0.0685i)11-s + (−0.505 − 0.659i)13-s + (−0.524 + 1.24i)14-s + (0.855 + 0.517i)16-s − 0.423·17-s + (1.19 + 0.494i)19-s + (−0.315 + 0.739i)20-s + (−0.0791 − 0.0799i)22-s + (−1.38 + 0.370i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00109 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00109 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.00109 - 0.999i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.00109 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88243 + 1.88037i\)
\(L(\frac12)\) \(\approx\) \(1.88243 + 1.88037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.191i)T \)
3 \( 1 \)
good5 \( 1 + (0.234 - 1.78i)T + (-4.82 - 1.29i)T^{2} \)
7 \( 1 + (0.927 - 3.45i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.296 + 0.227i)T + (2.84 + 10.6i)T^{2} \)
13 \( 1 + (1.82 + 2.37i)T + (-3.36 + 12.5i)T^{2} \)
17 \( 1 + 1.74T + 17T^{2} \)
19 \( 1 + (-5.20 - 2.15i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (6.63 - 1.77i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.67 - 0.746i)T + (28.0 - 7.50i)T^{2} \)
31 \( 1 + (-2.71 - 1.56i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.845 - 2.04i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.247 + 0.923i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.07 + 9.22i)T + (-11.1 - 41.5i)T^{2} \)
47 \( 1 + (-6.77 + 3.91i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0924 + 0.223i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-11.5 - 1.52i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (0.273 + 2.07i)T + (-58.9 + 15.7i)T^{2} \)
67 \( 1 + (-9.30 - 12.1i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (-6.98 + 6.98i)T - 71iT^{2} \)
73 \( 1 + (-3.07 - 3.07i)T + 73iT^{2} \)
79 \( 1 + (6.74 + 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.59 - 0.473i)T + (80.1 - 21.4i)T^{2} \)
89 \( 1 + (8.06 + 8.06i)T + 89iT^{2} \)
97 \( 1 + (-3.42 - 5.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.48893926626724761688683393408, −9.709211614391912702867384016452, −8.536704881752077923321244300105, −7.57087993422106131615110409137, −6.83887020607673905019627262913, −5.71540321756819336326549141259, −5.42769676140191625967254258243, −3.91375518935382032826566529075, −2.97994045765492048492202768725, −2.19209941789013296696715519225, 0.949328677358689540300431993384, 2.45109371825368122303505603584, 3.87030438738669361630986250735, 4.40170737206044434832512896083, 5.33931892892000261444382662306, 6.45490319577320879114368982613, 7.25559523162208741355568278406, 7.971165786251156931126468765179, 9.381821546870661852513143521530, 9.994837854026850329742298055832

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