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Properties

Label 2-6e3-72.13-c1-0-4
Degree $2$
Conductor $216$
Sign $0.591 + 0.806i$
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  + (−0.587 − 1.28i)2-s + (−1.30 + 1.51i)4-s + (1.97 + 1.14i)5-s + (−0.907 − 1.57i)7-s + (2.71 + 0.795i)8-s + (0.306 − 3.21i)10-s + (4.24 − 2.44i)11-s + (4.00 + 2.31i)13-s + (−1.48 + 2.09i)14-s + (−0.570 − 3.95i)16-s − 1.92·17-s − 2.12i·19-s + (−4.31 + 1.49i)20-s + (−5.64 − 4.01i)22-s + (1.15 − 2.00i)23-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (0.883 + 0.510i)5-s + (−0.343 − 0.594i)7-s + (0.959 + 0.281i)8-s + (0.0968 − 1.01i)10-s + (1.27 − 0.738i)11-s + (1.11 + 0.641i)13-s + (−0.398 + 0.559i)14-s + (−0.142 − 0.989i)16-s − 0.467·17-s − 0.488i·19-s + (−0.963 + 0.333i)20-s + (−1.20 − 0.856i)22-s + (0.241 − 0.418i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.591 + 0.806i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.591 + 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.988494 - 0.500806i\)
\(L(\frac12)\) \(\approx\) \(0.988494 - 0.500806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 1.28i)T \)
3 \( 1 \)
good5 \( 1 + (-1.97 - 1.14i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.907 + 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.24 + 2.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.00 - 2.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.92T + 17T^{2} \)
19 \( 1 + 2.12iT - 19T^{2} \)
23 \( 1 + (-1.15 + 2.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.16 - 1.82i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.65 - 4.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.98iT - 37T^{2} \)
41 \( 1 + (-2.36 + 4.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.20 - 1.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.02 - 3.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.95iT - 53T^{2} \)
59 \( 1 + (3.05 + 1.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.71 + 0.991i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.72 + 4.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + (4.97 + 8.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.12 - 1.80i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.49T + 89T^{2} \)
97 \( 1 + (-6.99 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.93596476382674533715014640758, −11.04200008453435366562625091106, −10.39733493435162784640347174934, −9.248663156593675723987022677898, −8.709318544069100935890792424054, −7.03924990487735367660133348263, −6.13645033717562492954598613792, −4.26775952685585280630932878647, −3.15126917820634993782343522014, −1.47155338009370854775886172198, 1.62023980263603489298278004554, 4.07034655654956497950688022404, 5.59691343206768738159567763153, 6.14387982901658151150248947095, 7.37536979820600018565946107606, 8.753622947563625340084897388665, 9.264102306246649224886055572838, 10.10612315566770780682802461679, 11.42698394432921080041412769992, 12.81850780119114029809602468874

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