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Properties

Label 2-864-288.13-c1-0-28
Degree $2$
Conductor $864$
Sign $0.980 - 0.198i$
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  + (−1.34 + 0.432i)2-s + (1.62 − 1.16i)4-s + (−0.328 + 2.49i)5-s + (2.51 + 0.674i)7-s + (−1.68 + 2.27i)8-s + (−0.637 − 3.50i)10-s + (2.79 − 3.63i)11-s + (4.24 − 3.26i)13-s + (−3.68 + 0.180i)14-s + (1.28 − 3.78i)16-s − 5.99i·17-s + (−3.23 − 1.33i)19-s + (2.37 + 4.44i)20-s + (−2.18 + 6.10i)22-s + (−2.38 + 0.639i)23-s + ⋯
L(s)  = 1  + (−0.952 + 0.305i)2-s + (0.812 − 0.582i)4-s + (−0.147 + 1.11i)5-s + (0.951 + 0.254i)7-s + (−0.595 + 0.803i)8-s + (−0.201 − 1.10i)10-s + (0.842 − 1.09i)11-s + (1.17 − 0.904i)13-s + (−0.983 + 0.0483i)14-s + (0.321 − 0.946i)16-s − 1.45i·17-s + (−0.741 − 0.307i)19-s + (0.531 + 0.993i)20-s + (−0.466 + 1.30i)22-s + (−0.498 + 0.133i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.980 - 0.198i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18990 + 0.119451i\)
\(L(\frac12)\) \(\approx\) \(1.18990 + 0.119451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.432i)T \)
3 \( 1 \)
good5 \( 1 + (0.328 - 2.49i)T + (-4.82 - 1.29i)T^{2} \)
7 \( 1 + (-2.51 - 0.674i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.79 + 3.63i)T + (-2.84 - 10.6i)T^{2} \)
13 \( 1 + (-4.24 + 3.26i)T + (3.36 - 12.5i)T^{2} \)
17 \( 1 + 5.99iT - 17T^{2} \)
19 \( 1 + (3.23 + 1.33i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.38 - 0.639i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.32 + 0.833i)T + (28.0 - 7.50i)T^{2} \)
31 \( 1 + (1.31 - 2.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.44 - 2.25i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-6.11 + 1.63i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.517 - 0.674i)T + (-11.1 - 41.5i)T^{2} \)
47 \( 1 + (-4.93 + 2.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.42 - 5.84i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.04 + 7.91i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-1.61 + 0.213i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (7.34 + 9.56i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (0.576 - 0.576i)T - 71iT^{2} \)
73 \( 1 + (-8.62 - 8.62i)T + 73iT^{2} \)
79 \( 1 + (-8.56 + 4.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.153 - 1.16i)T + (-80.1 + 21.4i)T^{2} \)
89 \( 1 + (12.1 - 12.1i)T - 89iT^{2} \)
97 \( 1 + (-7.91 - 13.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.35307080993194594554698429522, −9.139400775463243723919317905493, −8.490426600811249683557647800984, −7.79447467040421897466026718720, −6.78093122298683412987734906430, −6.16869451116072821129929342488, −5.15697919241965861063590598151, −3.50808972743084088046914626414, −2.51346216238249519981756737223, −0.970605209445094814533074891807, 1.29346469071124040281566164054, 1.88980952623757553012161545332, 3.99853030995043409086396029363, 4.39044868415189242918205235208, 5.98670115790743878054868488801, 6.86086661350165770294897927997, 7.955375690968832840389121817363, 8.605302917329649462918793265134, 9.039137725470536101665981449614, 10.13715600044190104816778989519

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