Properties

Label 2-6e3-216.85-c1-0-21
Degree $2$
Conductor $216$
Sign $0.863 + 0.504i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
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  + (−0.959 + 1.03i)2-s + (0.643 − 1.60i)3-s + (−0.159 − 1.99i)4-s + (0.866 + 0.152i)5-s + (1.05 + 2.21i)6-s + (−1.10 + 0.924i)7-s + (2.22 + 1.74i)8-s + (−2.17 − 2.07i)9-s + (−0.989 + 0.753i)10-s + (6.03 − 1.06i)11-s + (−3.30 − 1.02i)12-s + (−2.10 − 5.77i)13-s + (0.0963 − 2.03i)14-s + (0.803 − 1.29i)15-s + (−3.94 + 0.635i)16-s + (0.643 − 1.11i)17-s + ⋯
L(s)  = 1  + (−0.678 + 0.734i)2-s + (0.371 − 0.928i)3-s + (−0.0796 − 0.996i)4-s + (0.387 + 0.0683i)5-s + (0.429 + 0.902i)6-s + (−0.416 + 0.349i)7-s + (0.786 + 0.617i)8-s + (−0.723 − 0.690i)9-s + (−0.312 + 0.238i)10-s + (1.81 − 0.320i)11-s + (−0.955 − 0.296i)12-s + (−0.582 − 1.60i)13-s + (0.0257 − 0.543i)14-s + (0.207 − 0.334i)15-s + (−0.987 + 0.158i)16-s + (0.155 − 0.270i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.863 + 0.504i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.863 + 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980794 - 0.265381i\)
\(L(\frac12)\) \(\approx\) \(0.980794 - 0.265381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 - 1.03i)T \)
3 \( 1 + (-0.643 + 1.60i)T \)
good5 \( 1 + (-0.866 - 0.152i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.10 - 0.924i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-6.03 + 1.06i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.10 + 5.77i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.643 + 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.02 + 2.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.35 - 1.13i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.98 - 8.21i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.99 - 4.19i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-3.66 - 2.11i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.25 - 0.455i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (3.73 - 0.658i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (3.37 - 2.83i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 1.79iT - 53T^{2} \)
59 \( 1 + (-3.57 - 0.630i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (5.77 + 6.88i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.22 - 6.11i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.87 - 8.44i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.531 - 0.921i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.22 - 1.53i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.85 - 5.09i)T + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (-4.01 - 6.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.649 + 3.68i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.28908120569009207802586899612, −11.30208557071390110459991281848, −9.839442252987614087247773514758, −9.182735691508304559015406194593, −8.212551970961361572025701800456, −7.15533982425462581932016733136, −6.34801514195491334576024122825, −5.37348011710445629040740568358, −3.05987954974812021142015610193, −1.19566550210197877310928162311, 1.95487161218599116108963474161, 3.66970326855608581722127293801, 4.40436253451170454037642292387, 6.37596758871356026702891146102, 7.63580235648499313334421203248, 9.007560176905868785144699351084, 9.575521289990828900618044840787, 10.05218771049778899936878188278, 11.58972368286992136073884816880, 11.82074625181788096693514393046

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