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Properties

Label 2-864-216.59-c1-0-11
Degree $2$
Conductor $864$
Sign $0.899 + 0.437i$
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  + (−1.59 + 0.669i)3-s + (−3.00 + 1.09i)5-s + (−4.58 + 0.808i)7-s + (2.10 − 2.13i)9-s + (0.303 − 0.833i)11-s + (−1.22 + 1.45i)13-s + (4.07 − 3.76i)15-s + (−4.03 − 2.32i)17-s + (0.171 + 0.296i)19-s + (6.78 − 4.36i)21-s + (−1.00 + 5.68i)23-s + (4.02 − 3.37i)25-s + (−1.92 + 4.82i)27-s + (4.16 − 3.49i)29-s + (7.80 + 1.37i)31-s + ⋯
L(s)  = 1  + (−0.922 + 0.386i)3-s + (−1.34 + 0.489i)5-s + (−1.73 + 0.305i)7-s + (0.700 − 0.713i)9-s + (0.0914 − 0.251i)11-s + (−0.338 + 0.403i)13-s + (1.05 − 0.972i)15-s + (−0.978 − 0.564i)17-s + (0.0393 + 0.0680i)19-s + (1.48 − 0.952i)21-s + (−0.208 + 1.18i)23-s + (0.805 − 0.675i)25-s + (−0.370 + 0.928i)27-s + (0.774 − 0.649i)29-s + (1.40 + 0.247i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.899 + 0.437i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332960 - 0.0766379i\)
\(L(\frac12)\) \(\approx\) \(0.332960 - 0.0766379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 0.669i)T \)
good5 \( 1 + (3.00 - 1.09i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (4.58 - 0.808i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.303 + 0.833i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (1.22 - 1.45i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (4.03 + 2.32i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.171 - 0.296i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.00 - 5.68i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-4.16 + 3.49i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-7.80 - 1.37i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-2.31 - 1.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0346 + 0.0413i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.85 - 1.04i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.90 + 10.7i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 + (2.43 + 6.68i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.19 - 1.44i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-6.31 - 5.29i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.186 - 0.323i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.29 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.54 + 3.03i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (8.90 + 10.6i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (6.35 - 3.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.833 + 0.303i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.01921253693009816494547653233, −9.559419640108404865878224592167, −8.470386142819397066212872665698, −7.19944710485708811561724425575, −6.69644190579017361441916167802, −5.90211753081617011973838511130, −4.59854300539753430825588440003, −3.75616862799326545870564152472, −2.89061668737564456830754844706, −0.31634820642865051047318941236, 0.69530746790822388930858970496, 2.80326621415709000935692126203, 4.10026667654640654496891634633, 4.70078161915483403843933774979, 6.16841879133641449235712948730, 6.66493974519181127533137934661, 7.55634059231386409673567159711, 8.396550552917914391659889529642, 9.485081961394725555321069279230, 10.39015704375584505798976607163

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