Properties

Label 2-6e3-216.13-c1-0-22
Degree $2$
Conductor $216$
Sign $0.190 + 0.981i$
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  + (−1.23 − 0.689i)2-s + (1.71 − 0.252i)3-s + (1.04 + 1.70i)4-s + (2.05 − 2.45i)5-s + (−2.29 − 0.868i)6-s + (−4.54 − 1.65i)7-s + (−0.122 − 2.82i)8-s + (2.87 − 0.866i)9-s + (−4.23 + 1.61i)10-s + (0.638 + 0.761i)11-s + (2.22 + 2.65i)12-s + (1.85 + 0.326i)13-s + (4.46 + 5.16i)14-s + (2.90 − 4.72i)15-s + (−1.79 + 3.57i)16-s + (−0.690 + 1.19i)17-s + ⋯
L(s)  = 1  + (−0.873 − 0.487i)2-s + (0.989 − 0.145i)3-s + (0.524 + 0.851i)4-s + (0.920 − 1.09i)5-s + (−0.934 − 0.354i)6-s + (−1.71 − 0.624i)7-s + (−0.0434 − 0.999i)8-s + (0.957 − 0.288i)9-s + (−1.33 + 0.509i)10-s + (0.192 + 0.229i)11-s + (0.643 + 0.765i)12-s + (0.514 + 0.0906i)13-s + (1.19 + 1.38i)14-s + (0.750 − 1.21i)15-s + (−0.448 + 0.893i)16-s + (−0.167 + 0.290i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.190 + 0.981i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.876399 - 0.722408i\)
\(L(\frac12)\) \(\approx\) \(0.876399 - 0.722408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 + 0.689i)T \)
3 \( 1 + (-1.71 + 0.252i)T \)
good5 \( 1 + (-2.05 + 2.45i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (4.54 + 1.65i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.638 - 0.761i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.85 - 0.326i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.690 - 1.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.81 - 1.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.79 + 2.47i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.51 - 0.267i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (7.70 - 2.80i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-6.22 - 3.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0260 - 0.147i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-7.04 - 8.39i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (4.64 + 1.69i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 2.04iT - 53T^{2} \)
59 \( 1 + (0.104 - 0.124i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.77 - 7.63i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.85 + 0.502i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (6.18 - 10.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.83 + 4.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.784 - 4.45i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.477 - 0.0842i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-1.01 - 1.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.24 - 5.24i)T + (16.8 - 95.5i)T^{2} \)
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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

−12.54690287893675449439911642449, −10.75305867346958916786160168279, −9.758319379852072632656832883532, −9.276358143314633134710207431181, −8.574903045749590451721493825892, −7.20325876206780045860483007015, −6.28221294795083664864276312879, −4.12033737356792089324180687347, −2.90018259000409847106608415547, −1.30623912855939133225928603534, 2.32571648570850285303613826884, 3.28068746438141522611972486085, 5.77043784575498663765781423135, 6.59871358693524511553033165030, 7.40439402241891717029590998489, 9.128114642162554669583304924398, 9.220544829712533782974918871983, 10.25133174038632325851789568367, 11.06260283390730539196527320133, 12.86618157170883707056053409885

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