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Properties

Label 2-864-27.4-c1-0-26
Degree $2$
Conductor $864$
Sign $-0.335 + 0.941i$
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  + (−1.57 − 0.713i)3-s + (0.394 − 2.23i)5-s + (3.88 − 3.25i)7-s + (1.98 + 2.25i)9-s + (−0.823 − 4.67i)11-s + (4.81 − 1.75i)13-s + (−2.21 + 3.25i)15-s + (−3.30 + 5.72i)17-s + (1.30 + 2.26i)19-s + (−8.45 + 2.37i)21-s + (0.771 + 0.647i)23-s + (−0.154 − 0.0561i)25-s + (−1.52 − 4.96i)27-s + (4.01 + 1.46i)29-s + (−2.07 − 1.73i)31-s + ⋯
L(s)  = 1  + (−0.911 − 0.411i)3-s + (0.176 − 1.00i)5-s + (1.46 − 1.23i)7-s + (0.660 + 0.750i)9-s + (−0.248 − 1.40i)11-s + (1.33 − 0.486i)13-s + (−0.573 + 0.839i)15-s + (−0.801 + 1.38i)17-s + (0.299 + 0.518i)19-s + (−1.84 + 0.517i)21-s + (0.160 + 0.135i)23-s + (−0.0308 − 0.0112i)25-s + (−0.292 − 0.956i)27-s + (0.745 + 0.271i)29-s + (−0.371 − 0.312i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.335 + 0.941i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.335 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828176 - 1.17467i\)
\(L(\frac12)\) \(\approx\) \(0.828176 - 1.17467i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 + 0.713i)T \)
good5 \( 1 + (-0.394 + 2.23i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-3.88 + 3.25i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.823 + 4.67i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-4.81 + 1.75i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (3.30 - 5.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.30 - 2.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.771 - 0.647i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-4.01 - 1.46i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.07 + 1.73i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-0.217 + 0.376i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.16 + 1.51i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.751 + 4.25i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.238 + 0.200i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (0.632 - 3.58i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (3.75 - 3.15i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (13.0 - 4.73i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (2.24 - 3.89i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.59 - 7.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.13 - 0.776i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-8.21 - 2.98i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (4.16 + 7.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.71 + 9.74i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.33079408854370632135882942430, −8.653958942239188796359475813955, −8.307392838866714180308307983626, −7.46676741576394131914512365966, −6.20587669437042131202290400731, −5.56196484212083672365233349032, −4.63575316563031879545344973747, −3.79219417897199427901783831444, −1.52891850381191010494165527416, −0.911217914384978050755153864012, 1.71103389059054601074887502951, 2.84823275679047627202090658045, 4.57406921494727384015858676528, 4.93578389756042512359622801488, 6.11158797189140684940009657867, 6.81391519731563881105914286582, 7.75501213738785504575738691123, 8.978103467803449194058294996475, 9.551077680107692299905433271869, 10.78978087706932345371399586285

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