Properties

Label 2-6e3-216.13-c1-0-23
Degree $2$
Conductor $216$
Sign $0.749 + 0.662i$
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  + (1.15 − 0.820i)2-s + (−0.411 + 1.68i)3-s + (0.652 − 1.89i)4-s + (2.15 − 2.57i)5-s + (0.907 + 2.27i)6-s + (−1.78 − 0.649i)7-s + (−0.799 − 2.71i)8-s + (−2.66 − 1.38i)9-s + (0.374 − 4.73i)10-s + (3.21 + 3.83i)11-s + (2.91 + 1.87i)12-s + (3.61 + 0.637i)13-s + (−2.58 + 0.716i)14-s + (3.44 + 4.69i)15-s + (−3.14 − 2.46i)16-s + (−1.74 + 3.02i)17-s + ⋯
L(s)  = 1  + (0.814 − 0.580i)2-s + (−0.237 + 0.971i)3-s + (0.326 − 0.945i)4-s + (0.965 − 1.15i)5-s + (0.370 + 0.928i)6-s + (−0.674 − 0.245i)7-s + (−0.282 − 0.959i)8-s + (−0.887 − 0.461i)9-s + (0.118 − 1.49i)10-s + (0.970 + 1.15i)11-s + (0.840 + 0.541i)12-s + (1.00 + 0.176i)13-s + (−0.691 + 0.191i)14-s + (0.888 + 1.21i)15-s + (−0.786 − 0.617i)16-s + (−0.422 + 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73760 - 0.657545i\)
\(L(\frac12)\) \(\approx\) \(1.73760 - 0.657545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.820i)T \)
3 \( 1 + (0.411 - 1.68i)T \)
good5 \( 1 + (-2.15 + 2.57i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.78 + 0.649i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-3.21 - 3.83i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-3.61 - 0.637i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.74 - 3.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.73 - 1.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.93 - 1.06i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (6.61 - 1.16i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.842 + 0.306i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-6.21 - 3.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.341 - 1.93i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (5.36 + 6.39i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-2.63 - 0.959i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 9.52iT - 53T^{2} \)
59 \( 1 + (6.13 - 7.31i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.92 + 8.03i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-13.7 - 2.42i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.50 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.472 - 0.818i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.803 + 4.55i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.12 + 0.375i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (7.83 + 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.68 + 1.41i)T + (16.8 - 95.5i)T^{2} \)
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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

−12.35932167514805914515382973077, −11.28978985170902651673176461885, −10.16935452854548829736223282941, −9.578936234723485091997658199862, −8.809907705668471908653836021437, −6.46122576258760200451667591969, −5.77301334042209487836274717235, −4.52895123586952046308285375523, −3.79277995976669092246348505290, −1.72495880263088828855885168032, 2.38754099649589315717441900747, 3.51889310440156604753728448203, 5.67990266044452854445287954259, 6.35957153556649579833838425882, 6.76767676603450028940795837017, 8.188046134563859696914014875017, 9.306583729098705201214989384183, 11.01454723303839621744894487209, 11.45486389423883098002040406739, 12.80155282567234343001448450714

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