Properties

Label 2-864-32.13-c1-0-41
Degree $2$
Conductor $864$
Sign $-0.223 + 0.974i$
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  + (−0.322 − 1.37i)2-s + (−1.79 + 0.887i)4-s + (1.57 − 0.650i)5-s + (3.07 − 3.07i)7-s + (1.79 + 2.18i)8-s + (−1.40 − 1.95i)10-s + (−1.10 − 2.66i)11-s + (4.29 + 1.78i)13-s + (−5.22 − 3.24i)14-s + (2.42 − 3.18i)16-s + 5.78i·17-s + (1.92 + 0.798i)19-s + (−2.23 + 2.56i)20-s + (−3.32 + 2.38i)22-s + (−0.525 − 0.525i)23-s + ⋯
L(s)  = 1  + (−0.227 − 0.973i)2-s + (−0.896 + 0.443i)4-s + (0.702 − 0.291i)5-s + (1.16 − 1.16i)7-s + (0.636 + 0.771i)8-s + (−0.443 − 0.618i)10-s + (−0.333 − 0.805i)11-s + (1.19 + 0.493i)13-s + (−1.39 − 0.866i)14-s + (0.606 − 0.795i)16-s + 1.40i·17-s + (0.442 + 0.183i)19-s + (−0.500 + 0.572i)20-s + (−0.707 + 0.508i)22-s + (−0.109 − 0.109i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.223 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05341 - 1.32201i\)
\(L(\frac12)\) \(\approx\) \(1.05341 - 1.32201i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.322 + 1.37i)T \)
3 \( 1 \)
good5 \( 1 + (-1.57 + 0.650i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.07 + 3.07i)T - 7iT^{2} \)
11 \( 1 + (1.10 + 2.66i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-4.29 - 1.78i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 5.78iT - 17T^{2} \)
19 \( 1 + (-1.92 - 0.798i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.525 + 0.525i)T + 23iT^{2} \)
29 \( 1 + (-1.72 + 4.17i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.80T + 31T^{2} \)
37 \( 1 + (1.09 - 0.455i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (7.20 + 7.20i)T + 41iT^{2} \)
43 \( 1 + (-0.960 - 2.31i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 5.24iT - 47T^{2} \)
53 \( 1 + (0.00485 + 0.0117i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (8.86 - 3.67i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.439 + 1.06i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (1.17 - 2.84i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-7.54 + 7.54i)T - 71iT^{2} \)
73 \( 1 + (9.04 + 9.04i)T + 73iT^{2} \)
79 \( 1 + 8.85iT - 79T^{2} \)
83 \( 1 + (-13.0 - 5.41i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (12.4 - 12.4i)T - 89iT^{2} \)
97 \( 1 - 9.47T + 97T^{2} \)
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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

−10.29539120701185830909474474475, −9.117332999861378398030698397247, −8.311374992783474637851561926484, −7.83117560914399741640046446222, −6.33936717173785579396994451220, −5.31542210434043887609856833048, −4.27660025024756665731207621271, −3.52574563852004662305560105489, −1.88327950488059200233587981547, −1.08680497081564522500038430697, 1.47986830727855889476909466275, 2.83482230504349716450135597064, 4.61790912532965366948274138686, 5.27272001348231074239170906193, 6.01575327928480536360886274727, 6.96006772045621937991665611094, 7.972627816802727841691234506787, 8.557923978281543131909399504836, 9.439708374212759225271913841984, 10.12295579467342800113030292496

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