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Properties

Label 2-6e3-72.59-c1-0-0
Degree $2$
Conductor $216$
Sign $-0.657 - 0.753i$
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  + (−1.37 − 0.322i)2-s + (1.79 + 0.887i)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 + (5.80 − 1.75i)14-s + (2.42 + 3.18i)16-s + 1.49i·17-s − 3.42·19-s + (−1.88 + 1.25i)20-s + (1.57 − 0.477i)22-s + (−3.85 + 6.68i)23-s + ⋯
L(s)  = 1  + (−0.973 − 0.227i)2-s + (0.896 + 0.443i)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 + (1.55 − 0.469i)14-s + (0.606 + 0.795i)16-s + 0.362i·17-s − 0.785·19-s + (−0.420 + 0.280i)20-s + (0.336 − 0.101i)22-s + (−0.804 + 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.657 - 0.753i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135233 + 0.297551i\)
\(L(\frac12)\) \(\approx\) \(0.135233 + 0.297551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.322i)T \)
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

−12.45354466270516143243729894997, −11.68076818220123623134392175687, −10.46478282911438502704250519300, −9.791420581636471845475533993904, −8.903832411966953689408697929637, −7.74140437967944513728686149072, −6.76382735153459203869521590586, −5.74719754076435933059156939602, −3.51774927401439645416663223436, −2.43703024952855474369778669602, 0.34582365577185995921084818390, 2.68666844374878534197102742140, 4.41237398230841438932136066917, 6.17076559993628363937682734626, 6.93375710633730539137210776382, 8.027170889590176134985816613360, 9.054962106784999537510403517059, 10.01133258840713706514915916851, 10.59726022370264349556579437140, 11.96024151088687547774808694276

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