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Properties

Label 2-864-27.4-c1-0-9
Degree $2$
Conductor $864$
Sign $0.151 - 0.988i$
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 + 1.09i)3-s + (0.181 − 1.03i)5-s + (−0.246 + 0.206i)7-s + (0.591 − 2.94i)9-s + (0.447 + 2.53i)11-s + (−1.61 + 0.588i)13-s + (0.888 + 1.58i)15-s + (0.747 − 1.29i)17-s + (1.58 + 2.73i)19-s + (0.103 − 0.547i)21-s + (1.68 + 1.41i)23-s + (3.66 + 1.33i)25-s + (2.43 + 4.59i)27-s + (1.03 + 0.377i)29-s + (−1.18 − 0.990i)31-s + ⋯
L(s)  = 1  + (−0.773 + 0.633i)3-s + (0.0813 − 0.461i)5-s + (−0.0931 + 0.0781i)7-s + (0.197 − 0.980i)9-s + (0.134 + 0.765i)11-s + (−0.448 + 0.163i)13-s + (0.229 + 0.408i)15-s + (0.181 − 0.314i)17-s + (0.362 + 0.628i)19-s + (0.0225 − 0.119i)21-s + (0.351 + 0.294i)23-s + (0.733 + 0.266i)25-s + (0.468 + 0.883i)27-s + (0.192 + 0.0700i)29-s + (−0.212 − 0.177i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.151 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788439 + 0.676559i\)
\(L(\frac12)\) \(\approx\) \(0.788439 + 0.676559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 1.09i)T \)
good5 \( 1 + (-0.181 + 1.03i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.246 - 0.206i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.447 - 2.53i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (1.61 - 0.588i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.747 + 1.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.58 - 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.68 - 1.41i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-1.03 - 0.377i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.18 + 0.990i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (3.26 - 5.65i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.73 + 1.35i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.52 - 8.66i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.47 - 3.75i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 7.88T + 53T^{2} \)
59 \( 1 + (1.63 - 9.27i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (6.77 - 5.68i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (6.44 - 2.34i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-6.30 + 10.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.92 - 5.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.69 - 1.70i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.703 + 0.255i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-2.09 - 3.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.759 - 4.30i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.28591630885475101788800912393, −9.551325758911144111993666319803, −9.004801344248570387953108557185, −7.72197615551872148048267841073, −6.83571212157346991487280344128, −5.85539363271380969377305711344, −4.97644261288129850406686949390, −4.31748510601685292000490054842, −3.03675323408150554593985401168, −1.28089496442002097828021674832, 0.62609045353995317511462046591, 2.21633465257896870862021031049, 3.42500441625944390069410937539, 4.85129395815350802563790888699, 5.66681523458507112632334244907, 6.60880663296714213756544454783, 7.18661678463601373692524089234, 8.162576922191449713902982467948, 9.097668755324547575672832103544, 10.28475622038673752407572775147

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