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Properties

Label 2-864-36.23-c1-0-1
Degree $2$
Conductor $864$
Sign $-0.779 - 0.626i$
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  + (−1.81 − 1.04i)5-s + (−0.143 + 0.0829i)7-s + (−0.784 − 1.35i)11-s + (−1.93 + 3.34i)13-s − 5.27i·17-s + 8.05i·19-s + (−2.67 + 4.63i)23-s + (−0.298 − 0.516i)25-s + (−6.75 + 3.89i)29-s + (2.10 + 1.21i)31-s + 0.348·35-s − 8.53·37-s + (−2.47 − 1.43i)41-s + (3.42 − 1.97i)43-s + (3.68 + 6.38i)47-s + ⋯
L(s)  = 1  + (−0.812 − 0.469i)5-s + (−0.0543 + 0.0313i)7-s + (−0.236 − 0.409i)11-s + (−0.535 + 0.928i)13-s − 1.27i·17-s + 1.84i·19-s + (−0.557 + 0.966i)23-s + (−0.0596 − 0.103i)25-s + (−1.25 + 0.724i)29-s + (0.378 + 0.218i)31-s + 0.0588·35-s − 1.40·37-s + (−0.387 − 0.223i)41-s + (0.521 − 0.301i)43-s + (0.537 + 0.931i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.779 - 0.626i$
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: \((2,\ 864,\ (\ :1/2),\ -0.779 - 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105313 + 0.299234i\)
\(L(\frac12)\) \(\approx\) \(0.105313 + 0.299234i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 + 1.04i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.143 - 0.0829i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.784 + 1.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.93 - 3.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.27iT - 17T^{2} \)
19 \( 1 - 8.05iT - 19T^{2} \)
23 \( 1 + (2.67 - 4.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.75 - 3.89i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.10 - 1.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.53T + 37T^{2} \)
41 \( 1 + (2.47 + 1.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.42 + 1.97i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.68 - 6.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.40iT - 53T^{2} \)
59 \( 1 + (5.49 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.11 + 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.45 + 0.841i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + (-6.31 + 3.64i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.35 + 2.34i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.40iT - 89T^{2} \)
97 \( 1 + (-0.903 - 1.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49842010445527831352367182617, −9.531582128948814027140478586649, −8.870820845770140513591561393367, −7.80239111756923778786327859224, −7.36606040775599272071971489336, −6.09894241613494444020858137405, −5.14959670845423032072405401780, −4.17020467927394415253348141636, −3.25956506762951283826185106601, −1.69490276380711529514913975513, 0.14756751368874980584200165406, 2.23905862412219906545135829185, 3.36573453048972755788362049924, 4.34875841388226733125200492276, 5.36471997656578037283028128174, 6.51087498946561648844261005123, 7.34986412807949759451908757436, 8.025774883285868536600153889896, 8.908811877627962813788740897536, 10.00972917829112270944335879022

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