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Properties

Label 2-6e3-72.11-c1-0-3
Degree $2$
Conductor $216$
Sign $0.864 + 0.501i$
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  + (−0.409 − 1.35i)2-s + (−1.66 + 1.10i)4-s + (0.565 + 0.978i)5-s + (3.71 + 2.14i)7-s + (2.18 + 1.79i)8-s + (1.09 − 1.16i)10-s + (−1.00 − 0.582i)11-s + (2.64 − 1.52i)13-s + (1.38 − 5.90i)14-s + (1.54 − 3.69i)16-s − 1.49i·17-s − 3.42·19-s + (−2.02 − 1.00i)20-s + (−0.375 + 1.60i)22-s + (3.85 + 6.68i)23-s + ⋯
L(s)  = 1  + (−0.289 − 0.957i)2-s + (−0.832 + 0.554i)4-s + (0.252 + 0.437i)5-s + (1.40 + 0.810i)7-s + (0.771 + 0.636i)8-s + (0.345 − 0.368i)10-s + (−0.304 − 0.175i)11-s + (0.733 − 0.423i)13-s + (0.369 − 1.57i)14-s + (0.385 − 0.922i)16-s − 0.362i·17-s − 0.785·19-s + (−0.453 − 0.224i)20-s + (−0.0800 + 0.342i)22-s + (0.804 + 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.864 + 0.501i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.864 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10611 - 0.297744i\)
\(L(\frac12)\) \(\approx\) \(1.10611 - 0.297744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.409 + 1.35i)T \)
3 \( 1 \)
good5 \( 1 + (-0.565 - 0.978i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.71 - 2.14i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.00 + 0.582i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.64 + 1.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.49iT - 17T^{2} \)
19 \( 1 + 3.42T + 19T^{2} \)
23 \( 1 + (-3.85 - 6.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.709 + 1.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.66 - 2.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.97iT - 37T^{2} \)
41 \( 1 + (-4.23 + 2.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.74 - 3.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.77 + 3.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + (7.50 - 4.33i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 + 1.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.58 + 9.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.54T + 71T^{2} \)
73 \( 1 + 7.06T + 73T^{2} \)
79 \( 1 + (2.24 + 1.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.98 - 2.30i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.63iT - 89T^{2} \)
97 \( 1 + (-3.35 + 5.81i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.03111730239518742671216500053, −11.08511798355722397933734210418, −10.68239095886623010006178728057, −9.280881122274240949430483249023, −8.485064223077853151616149694203, −7.56668826538837312327926476707, −5.74504864881787360768765295622, −4.67380923624221462943428937905, −3.06685401357228604406068378300, −1.74108316832948756750271787310, 1.41099012573107542733979138370, 4.25866998968054581710550401523, 5.01752049296537929890927532854, 6.33530516392561881433843256596, 7.46648498827178383021911554640, 8.363508287636607257335706704934, 9.107671669085552429526663340438, 10.51668835564347271957334462639, 11.07661905103760503326306544584, 12.72912842620429275246405241993

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