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Properties

Label 2-864-108.11-c1-0-5
Degree $2$
Conductor $864$
Sign $0.194 - 0.980i$
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  + (−1.71 + 0.272i)3-s + (0.354 − 0.975i)5-s + (−4.63 − 0.817i)7-s + (2.85 − 0.930i)9-s + (2.32 − 0.846i)11-s + (−2.86 + 2.40i)13-s + (−0.341 + 1.76i)15-s + (2.82 − 1.62i)17-s + (−3.42 − 1.97i)19-s + (8.15 + 0.137i)21-s + (1.06 + 6.04i)23-s + (3.00 + 2.52i)25-s + (−4.62 + 2.36i)27-s + (−5.10 + 6.07i)29-s + (10.2 − 1.81i)31-s + ⋯
L(s)  = 1  + (−0.987 + 0.157i)3-s + (0.158 − 0.436i)5-s + (−1.75 − 0.309i)7-s + (0.950 − 0.310i)9-s + (0.700 − 0.255i)11-s + (−0.794 + 0.666i)13-s + (−0.0882 + 0.455i)15-s + (0.684 − 0.394i)17-s + (−0.785 − 0.453i)19-s + (1.77 + 0.0299i)21-s + (0.222 + 1.25i)23-s + (0.600 + 0.504i)25-s + (−0.890 + 0.455i)27-s + (−0.947 + 1.12i)29-s + (1.84 − 0.325i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.194 - 0.980i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.194 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475277 + 0.390237i\)
\(L(\frac12)\) \(\approx\) \(0.475277 + 0.390237i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 0.272i)T \)
good5 \( 1 + (-0.354 + 0.975i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (4.63 + 0.817i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.32 + 0.846i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (2.86 - 2.40i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.82 + 1.62i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.42 + 1.97i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.06 - 6.04i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.10 - 6.07i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-10.2 + 1.81i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.0304 - 0.0527i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.26 + 5.08i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.68 - 10.1i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.20 - 6.80i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 6.06iT - 53T^{2} \)
59 \( 1 + (8.99 + 3.27i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.05 - 5.98i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.695 - 0.828i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-3.24 - 5.61i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.27 + 2.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.30 - 2.74i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-5.87 - 4.92i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-5.37 - 3.10i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.14 - 0.417i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.25718827765314232015453580783, −9.399315530718666320943478217860, −9.231509112167625654811187105903, −7.47627052705235248526128655768, −6.72919874426558713494550034563, −6.12068598031836826201465398591, −5.10195185892030563578272299381, −4.11465106445978035019302731082, −3.06298903707667887373776604048, −1.12926524143249601566806490259, 0.39280624851651728670848782536, 2.35607153162450387016119277972, 3.52683361753198089893685290657, 4.68953514469944497990420641434, 5.93390497723579931273804197688, 6.41795207924263918547478723816, 7.03542788002927671582217540153, 8.239352119285468435533119004371, 9.457206651816438628091598627291, 10.22540207843168012340207130585

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