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Properties

Label 2-6e3-24.11-c1-0-7
Degree $2$
Conductor $216$
Sign $0.487 - 0.873i$
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  + (1.08 + 0.903i)2-s + (0.366 + 1.96i)4-s + 1.59·5-s − 1.43i·7-s + (−1.37 + 2.46i)8-s + (1.73 + 1.43i)10-s + 4.93i·11-s − 5.37i·13-s + (1.30 − 1.56i)14-s + (−3.73 + 1.43i)16-s − 1.32i·17-s + 0.267·19-s + (0.582 + 3.13i)20-s + (−4.46 + 5.37i)22-s − 5.94·23-s + ⋯
L(s)  = 1  + (0.769 + 0.639i)2-s + (0.183 + 0.983i)4-s + 0.712·5-s − 0.544i·7-s + (−0.487 + 0.873i)8-s + (0.547 + 0.455i)10-s + 1.48i·11-s − 1.48i·13-s + (0.347 − 0.418i)14-s + (−0.933 + 0.359i)16-s − 0.320i·17-s + 0.0614·19-s + (0.130 + 0.700i)20-s + (−0.951 + 1.14i)22-s − 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.487 - 0.873i$
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: \((2,\ 216,\ (\ :1/2),\ 0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63874 + 0.961792i\)
\(L(\frac12)\) \(\approx\) \(1.63874 + 0.961792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 - 0.903i)T \)
3 \( 1 \)
good5 \( 1 - 1.59T + 5T^{2} \)
7 \( 1 + 1.43iT - 7T^{2} \)
11 \( 1 - 4.93iT - 11T^{2} \)
13 \( 1 + 5.37iT - 13T^{2} \)
17 \( 1 + 1.32iT - 17T^{2} \)
19 \( 1 - 0.267T + 19T^{2} \)
23 \( 1 + 5.94T + 23T^{2} \)
29 \( 1 - 8.70T + 29T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 - 2.49iT - 37T^{2} \)
41 \( 1 + 9.87iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 9.12T + 47T^{2} \)
53 \( 1 + 5.51T + 53T^{2} \)
59 \( 1 - 4.93iT - 59T^{2} \)
61 \( 1 - 5.37iT - 61T^{2} \)
67 \( 1 - 1.19T + 67T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 + 7.92T + 73T^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 - 7.23iT - 83T^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 - 9.39T + 97T^{2} \)
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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

−12.67212128694233668529950166147, −11.89118458798677799664844724906, −10.42034585512795390399259749432, −9.683463525305141614318582920687, −8.143745740818759717991930723822, −7.34782838128611224441648475077, −6.21816835300278862599291100121, −5.19422949737601980516745538247, −4.05295280620260160674676803954, −2.43469222275419838319908683899, 1.79704013317530939188108252714, 3.20986074434543347166592576019, 4.63147395194452936244809177611, 5.92388794560208938252506647934, 6.47958965165534209599223323481, 8.446201725634776638901120232130, 9.412627470516339988346649601765, 10.35677057646674308297685346393, 11.42291754707083102425770551628, 12.05057522720611625487355568586

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