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Properties

Label 2-6e3-24.11-c1-0-12
Degree $2$
Conductor $216$
Sign $0.166 + 0.986i$
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  + (0.637 − 1.26i)2-s + (−1.18 − 1.61i)4-s + 3.42·5-s − 0.505i·7-s + (−2.78 + 0.469i)8-s + (2.18 − 4.32i)10-s + 3.31i·11-s − 5.43i·13-s + (−0.637 − 0.322i)14-s + (−1.18 + 3.82i)16-s + 1.58i·17-s − 6.74·19-s + (−4.06 − 5.51i)20-s + (4.18 + 2.11i)22-s + 4.30·23-s + ⋯
L(s)  = 1  + (0.451 − 0.892i)2-s + (−0.593 − 0.805i)4-s + 1.53·5-s − 0.191i·7-s + (−0.986 + 0.166i)8-s + (0.691 − 1.36i)10-s + 1.00i·11-s − 1.50i·13-s + (−0.170 − 0.0861i)14-s + (−0.296 + 0.955i)16-s + 0.384i·17-s − 1.54·19-s + (−0.908 − 1.23i)20-s + (0.892 + 0.451i)22-s + 0.897·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.166 + 0.986i$
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: \((2,\ 216,\ (\ :1/2),\ 0.166 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28298 - 1.08492i\)
\(L(\frac12)\) \(\approx\) \(1.28298 - 1.08492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.637 + 1.26i)T \)
3 \( 1 \)
good5 \( 1 - 3.42T + 5T^{2} \)
7 \( 1 + 0.505iT - 7T^{2} \)
11 \( 1 - 3.31iT - 11T^{2} \)
13 \( 1 + 5.43iT - 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 + 6.74T + 19T^{2} \)
23 \( 1 - 4.30T + 23T^{2} \)
29 \( 1 + 2.55T + 29T^{2} \)
31 \( 1 - 4.92iT - 31T^{2} \)
37 \( 1 - 7.45iT - 37T^{2} \)
41 \( 1 - 8.51iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 5.10T + 47T^{2} \)
53 \( 1 + 0.875T + 53T^{2} \)
59 \( 1 + 6.63iT - 59T^{2} \)
61 \( 1 + 10.8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 + 3.74T + 73T^{2} \)
79 \( 1 + 6.44iT - 79T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 - 3.75iT - 89T^{2} \)
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

−12.45107684291295739227361017555, −10.87696194498750572020822484258, −10.27290571112059143507105095248, −9.582790703848586812126255059710, −8.439207320806334264955447242970, −6.64251357576350219775876209725, −5.62994459270790815644211709391, −4.65202205570239063286729143235, −2.92953664176158627376357050248, −1.67387876487883170690795925035, 2.37137446776201990643666290341, 4.16336413846382076806159682866, 5.53452491685405563692100455047, 6.21296835242234956019807558402, 7.17369544057194018534886649291, 8.919686391248654679338182873616, 9.063570898159216293278642163148, 10.52831879173803804876308295172, 11.74172864154345246498941715674, 12.96920347537960604743801951418

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