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Properties

Label 2-864-108.11-c1-0-15
Degree $2$
Conductor $864$
Sign $0.298 - 0.954i$
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  + (−1.38 + 1.04i)3-s + (−1.14 + 3.15i)5-s + (3.41 + 0.602i)7-s + (0.828 − 2.88i)9-s + (5.46 − 1.98i)11-s + (2.22 − 1.86i)13-s + (−1.69 − 5.55i)15-s + (1.37 − 0.795i)17-s + (4.19 + 2.42i)19-s + (−5.35 + 2.72i)21-s + (−0.571 − 3.24i)23-s + (−4.78 − 4.01i)25-s + (1.85 + 4.85i)27-s + (−6.50 + 7.74i)29-s + (8.83 − 1.55i)31-s + ⋯
L(s)  = 1  + (−0.798 + 0.601i)3-s + (−0.513 + 1.40i)5-s + (1.29 + 0.227i)7-s + (0.276 − 0.961i)9-s + (1.64 − 0.599i)11-s + (0.618 − 0.518i)13-s + (−0.438 − 1.43i)15-s + (0.334 − 0.192i)17-s + (0.963 + 0.556i)19-s + (−1.16 + 0.595i)21-s + (−0.119 − 0.675i)23-s + (−0.957 − 0.803i)25-s + (0.357 + 0.933i)27-s + (−1.20 + 1.43i)29-s + (1.58 − 0.279i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.298 - 0.954i$
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: \((2,\ 864,\ (\ :1/2),\ 0.298 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17978 + 0.867081i\)
\(L(\frac12)\) \(\approx\) \(1.17978 + 0.867081i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 1.04i)T \)
good5 \( 1 + (1.14 - 3.15i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-3.41 - 0.602i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-5.46 + 1.98i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-2.22 + 1.86i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.37 + 0.795i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.19 - 2.42i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.571 + 3.24i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (6.50 - 7.74i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-8.83 + 1.55i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.74 - 3.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.53 + 6.59i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.13 + 3.12i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.865 - 4.90i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 4.99iT - 53T^{2} \)
59 \( 1 + (2.57 + 0.938i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.47 + 8.34i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.88 + 8.20i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.30 - 2.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.241 + 0.418i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.268 - 0.320i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.52 + 1.27i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-12.3 - 7.10i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.14 - 2.96i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.72387947464888066772838301818, −9.648666713429660930621301342177, −8.663129368206653417562001361416, −7.72217243352850118332675727688, −6.74228700901482152826306837405, −6.04871521610005090695798833721, −5.05253003338883258852176027718, −3.88800826571018532646524857460, −3.23337851796208425706134759749, −1.28778399630060450158803827770, 1.08341349463746384417665685906, 1.62833163302501958061925140788, 4.06917892227223255943099354789, 4.61798035399722983979855825612, 5.48164465272144289656435168521, 6.54434978770991658269471265799, 7.55496760878816188760833260977, 8.176653144994335799570410326907, 9.043669813158482278487753750893, 9.919794127484789905904872911552

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