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Properties

Label 2-6e3-24.11-c1-0-3
Degree $2$
Conductor $216$
Sign $0.752 - 0.658i$
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  + (−1.35 − 0.396i)2-s + (1.68 + 1.07i)4-s + 0.505·5-s + 3.42i·7-s + (−1.86 − 2.12i)8-s + (−0.686 − 0.200i)10-s + 3.31i·11-s − 2.55i·13-s + (1.35 − 4.65i)14-s + (1.68 + 3.62i)16-s + 5.04i·17-s + 4.74·19-s + (0.852 + 0.543i)20-s + (1.31 − 4.50i)22-s + 6.44·23-s + ⋯
L(s)  = 1  + (−0.959 − 0.280i)2-s + (0.843 + 0.537i)4-s + 0.226·5-s + 1.29i·7-s + (−0.658 − 0.752i)8-s + (−0.216 − 0.0633i)10-s + 1.00i·11-s − 0.707i·13-s + (0.362 − 1.24i)14-s + (0.421 + 0.906i)16-s + 1.22i·17-s + 1.08·19-s + (0.190 + 0.121i)20-s + (0.280 − 0.959i)22-s + 1.34·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.752 - 0.658i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765045 + 0.287551i\)
\(L(\frac12)\) \(\approx\) \(0.765045 + 0.287551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.396i)T \)
3 \( 1 \)
good5 \( 1 - 0.505T + 5T^{2} \)
7 \( 1 - 3.42iT - 7T^{2} \)
11 \( 1 - 3.31iT - 11T^{2} \)
13 \( 1 + 2.55iT - 13T^{2} \)
17 \( 1 - 5.04iT - 17T^{2} \)
19 \( 1 - 4.74T + 19T^{2} \)
23 \( 1 - 6.44T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 - 5.97iT - 31T^{2} \)
37 \( 1 + 11.1iT - 37T^{2} \)
41 \( 1 + 1.87iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 5.93T + 53T^{2} \)
59 \( 1 + 6.63iT - 59T^{2} \)
61 \( 1 + 5.10iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 4.41T + 71T^{2} \)
73 \( 1 - 7.74T + 73T^{2} \)
79 \( 1 - 4.30iT - 79T^{2} \)
83 \( 1 + 3.61iT - 83T^{2} \)
89 \( 1 + 17.0iT - 89T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−12.38948342361773064743769524531, −11.39312952335440304553915688072, −10.32153051473164650974883437045, −9.485183224019188783806628085571, −8.629190705727872295047027217020, −7.64988805209780768577822403743, −6.43050250709880926884471565730, −5.24777264330101516492658395562, −3.20987585467554131140015515219, −1.86902757423373283855491132023, 1.02441338427157735853405668572, 3.10887027102564679464861558388, 4.94137065142625821867070671754, 6.38520551959799855303677001626, 7.23461574629368033201511942517, 8.182994374496439457076602851155, 9.392998899413629339908931316037, 10.04943902268457754151362567257, 11.20407988118635539971539909992, 11.69339888052578842264002961191

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