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Properties

Label 2-864-72.11-c1-0-3
Degree $2$
Conductor $864$
Sign $0.377 - 0.925i$
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  + (1.74 + 3.01i)5-s + (1.80 + 1.04i)7-s + (−0.116 − 0.0675i)11-s + (2.63 − 1.52i)13-s + 4.19i·17-s − 0.919·19-s + (−0.689 − 1.19i)23-s + (−3.57 + 6.19i)25-s + (4.24 − 7.34i)29-s + (−4.39 + 2.53i)31-s + 7.27i·35-s + 1.61i·37-s + (−1.79 + 1.03i)41-s + (−5.41 + 9.37i)43-s + (0.205 − 0.356i)47-s + ⋯
L(s)  = 1  + (0.779 + 1.35i)5-s + (0.683 + 0.394i)7-s + (−0.0352 − 0.0203i)11-s + (0.731 − 0.422i)13-s + 1.01i·17-s − 0.210·19-s + (−0.143 − 0.249i)23-s + (−0.715 + 1.23i)25-s + (0.787 − 1.36i)29-s + (−0.790 + 0.456i)31-s + 1.23i·35-s + 0.265i·37-s + (−0.280 + 0.161i)41-s + (−0.825 + 1.42i)43-s + (0.0300 − 0.0519i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.377 - 0.925i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.377 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59320 + 1.07064i\)
\(L(\frac12)\) \(\approx\) \(1.59320 + 1.07064i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.74 - 3.01i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.80 - 1.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.116 + 0.0675i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.63 + 1.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.19iT - 17T^{2} \)
19 \( 1 + 0.919T + 19T^{2} \)
23 \( 1 + (0.689 + 1.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.24 + 7.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.39 - 2.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.61iT - 37T^{2} \)
41 \( 1 + (1.79 - 1.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.41 - 9.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.205 + 0.356i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.968T + 53T^{2} \)
59 \( 1 + (-3.88 + 2.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.44 - 4.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.15 + 5.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 4.06T + 73T^{2} \)
79 \( 1 + (-10.8 - 6.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.23 - 3.02i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.35iT - 89T^{2} \)
97 \( 1 + (0.477 - 0.826i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.38814565920056373709364700298, −9.694400337766778996971116106602, −8.497488783745462755191547533907, −7.899375556678735145756121110922, −6.62466196069757021644277595980, −6.16319798210093468031506357123, −5.18733536428248953545360015008, −3.82771142374593056095344122860, −2.73817433122494892871409563518, −1.73128205486260862696723785920, 1.02327751074269719287344111031, 2.04351050154369492086973934525, 3.74950402852670135239512923327, 4.87487415773963321021594244066, 5.34044657899876722760448083652, 6.49819148489783565436172183421, 7.52824352687329009734477057803, 8.580490042093929908897473395976, 9.015781057668825876629651062174, 9.896176173585238334759773910138

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