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Properties

Label 2-6e3-24.11-c1-0-14
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

\(L(1)\) \(\approx\) \(1.88042 - 0.706778i\)
\(L(\frac12)\) \(\approx\) \(1.88042 - 0.706778i\)
\(L(\frac{3}{2})\) 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

−12.17096492742120167834741427934, −11.48378096183601546827425196288, −10.33873531827654432795428004197, −9.741084468291331062068063648746, −7.81550364213789775615067837790, −7.06682051644744674715130348383, −5.89034174113451245554688598712, −4.39230406659308347966542086800, −3.76881541038612020331620973699, −1.79268394713401547332541236107, 2.52858279256650663850279503411, 3.68191862269730974460727216701, 5.39682650992730834955377378622, 5.79928436647411051525819290146, 7.31329708181304976850481062841, 8.256928370679820060869553082796, 9.377559725556292276675915727860, 10.86738496714636717264764819110, 11.83577373533088134478643680279, 12.26640381820150103007422813901

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