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Properties

Label 2-6e3-8.5-c3-0-28
Degree $2$
Conductor $216$
Sign $0.937 - 0.348i$
Analytic cond. $12.7444$
Root an. cond. $3.56993$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.334 + 2.80i)2-s + (−7.77 + 1.88i)4-s − 6.44i·5-s + 8.32·7-s + (−7.88 − 21.2i)8-s + (18.1 − 2.16i)10-s + 23.3i·11-s − 77.9i·13-s + (2.78 + 23.3i)14-s + (56.9 − 29.2i)16-s + 83.4·17-s − 38.4i·19-s + (12.1 + 50.1i)20-s + (−65.5 + 7.81i)22-s + 135.·23-s + ⋯
L(s)  = 1  + (0.118 + 0.992i)2-s + (−0.971 + 0.235i)4-s − 0.576i·5-s + 0.449·7-s + (−0.348 − 0.937i)8-s + (0.572 − 0.0683i)10-s + 0.639i·11-s − 1.66i·13-s + (0.0532 + 0.446i)14-s + (0.889 − 0.457i)16-s + 1.19·17-s − 0.463i·19-s + (0.135 + 0.560i)20-s + (−0.635 + 0.0757i)22-s + 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.937 - 0.348i$
Analytic conductor: \(12.7444\)
Root analytic conductor: \(3.56993\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :3/2),\ 0.937 - 0.348i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.69722 + 0.305445i\)
\(L(\frac12)\) \(\approx\) \(1.69722 + 0.305445i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.334 - 2.80i)T \)
3 \( 1 \)
good5 \( 1 + 6.44iT - 125T^{2} \)
7 \( 1 - 8.32T + 343T^{2} \)
11 \( 1 - 23.3iT - 1.33e3T^{2} \)
13 \( 1 + 77.9iT - 2.19e3T^{2} \)
17 \( 1 - 83.4T + 4.91e3T^{2} \)
19 \( 1 + 38.4iT - 6.85e3T^{2} \)
23 \( 1 - 135.T + 1.21e4T^{2} \)
29 \( 1 - 150. iT - 2.43e4T^{2} \)
31 \( 1 - 6.02T + 2.97e4T^{2} \)
37 \( 1 + 141. iT - 5.06e4T^{2} \)
41 \( 1 + 179.T + 6.89e4T^{2} \)
43 \( 1 + 156. iT - 7.95e4T^{2} \)
47 \( 1 - 287.T + 1.03e5T^{2} \)
53 \( 1 + 399. iT - 1.48e5T^{2} \)
59 \( 1 - 473. iT - 2.05e5T^{2} \)
61 \( 1 + 382. iT - 2.26e5T^{2} \)
67 \( 1 + 735. iT - 3.00e5T^{2} \)
71 \( 1 - 842.T + 3.57e5T^{2} \)
73 \( 1 - 1.03e3T + 3.89e5T^{2} \)
79 \( 1 + 137.T + 4.93e5T^{2} \)
83 \( 1 + 443. iT - 5.71e5T^{2} \)
89 \( 1 + 1.42e3T + 7.04e5T^{2} \)
97 \( 1 + 1.21e3T + 9.12e5T^{2} \)
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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

−12.46521015643784270832056113407, −10.81738441095692129914990678411, −9.753345491131594248281813939907, −8.702125001125672772733589506792, −7.86998465586252152565150217269, −6.95388072621138347360961721370, −5.42528885241410897288152803317, −4.93041330224021812229201552363, −3.31710888121971960099758135629, −0.858339249824833809916944373982, 1.30417176476851816112720647194, 2.80729164279946315850518717806, 4.02382761750577836090014432144, 5.26569137168074800037265089040, 6.61365451837946879573262996502, 8.033292385063411421079019315420, 9.082082470775225694943517135106, 10.01822774258019719108571409396, 11.06994190154219337653869718965, 11.60531990520779405865879260459

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