Properties

Label 2-864-108.11-c1-0-0
Degree $2$
Conductor $864$
Sign $-0.954 - 0.298i$
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

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.0349562 + 0.228814i\)
\(L(\frac12)\) \(\approx\) \(0.0349562 + 0.228814i\)
\(L(\frac{3}{2})\) 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

−10.53747343622243846539975939227, −9.785033294277165636133187191091, −8.807462374715204605136424102386, −7.71893518817149464203833965220, −7.19779827355010365801213222074, −6.61917351410262250422873550256, −5.45775145114880939486518629318, −3.63091414722690871762115495881, −3.21492416838134853358828084672, −2.25017337218369933016670004480, 0.091967627478869971792054318679, 2.19643555623563097437028783034, 3.45908821716205892417812487195, 4.17966630097774030063287364039, 5.26765931163389081005100986542, 6.08696695437850280625126551933, 7.61848801145588513630746031912, 8.276792348341990307018682783082, 8.904721803050649628767587041329, 9.613725586936104090234474694114

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