Properties

Label 2-864-36.11-c1-0-4
Degree $2$
Conductor $864$
Sign $0.934 - 0.356i$
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  + (0.398 − 0.229i)5-s + (4.28 + 2.47i)7-s + (1.17 − 2.03i)11-s + (−0.0384 − 0.0666i)13-s + 5.92i·17-s − 3.59i·19-s + (−2.41 − 4.18i)23-s + (−2.39 + 4.14i)25-s + (6.54 + 3.78i)29-s + (0.663 − 0.383i)31-s + 2.27·35-s − 5.47·37-s + (0.986 − 0.569i)41-s + (6.79 + 3.92i)43-s + (3.01 − 5.21i)47-s + ⋯
L(s)  = 1  + (0.178 − 0.102i)5-s + (1.61 + 0.934i)7-s + (0.354 − 0.613i)11-s + (−0.0106 − 0.0184i)13-s + 1.43i·17-s − 0.824i·19-s + (−0.503 − 0.872i)23-s + (−0.478 + 0.829i)25-s + (1.21 + 0.702i)29-s + (0.119 − 0.0687i)31-s + 0.384·35-s − 0.900·37-s + (0.154 − 0.0889i)41-s + (1.03 + 0.597i)43-s + (0.439 − 0.760i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91609 + 0.353149i\)
\(L(\frac12)\) \(\approx\) \(1.91609 + 0.353149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.398 + 0.229i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-4.28 - 2.47i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.17 + 2.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0384 + 0.0666i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.92iT - 17T^{2} \)
19 \( 1 + 3.59iT - 19T^{2} \)
23 \( 1 + (2.41 + 4.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.54 - 3.78i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.663 + 0.383i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.47T + 37T^{2} \)
41 \( 1 + (-0.986 + 0.569i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.79 - 3.92i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.01 + 5.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.71iT - 53T^{2} \)
59 \( 1 + (-4.15 - 7.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.63 + 4.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.84 - 1.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 7.51T + 73T^{2} \)
79 \( 1 + (2.52 + 1.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.44 + 7.69i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.71iT - 89T^{2} \)
97 \( 1 + (-3.16 + 5.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.40145446449584681670807049887, −9.095404419481344778402321517268, −8.523697408811834841190915359799, −7.964369215058879791444298632985, −6.68280291531266148260745939706, −5.72197927033670830974264828432, −4.99475704804114654734710831781, −3.96037489851306499135018766318, −2.47897960969473914056389960951, −1.42562013671301621575863516197, 1.16864354102782050102085744516, 2.34337763878319064970385222165, 3.97993906033322572851645778897, 4.65801012692934491484330878881, 5.61431150681218473978125104463, 6.86511291524128619236942721344, 7.62966336971914026549426517954, 8.217713119085464179852613410830, 9.384066731483545861359204325361, 10.17887417259535705887806744107

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