Properties

Label 2-6e3-216.101-c2-0-5
Degree $2$
Conductor $216$
Sign $-0.882 - 0.470i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.96 + 0.361i)2-s + (1.70 + 2.46i)3-s + (3.73 − 1.42i)4-s + (−0.578 − 3.27i)5-s + (−4.24 − 4.23i)6-s + (−6.44 − 5.41i)7-s + (−6.84 + 4.14i)8-s + (−3.17 + 8.42i)9-s + (2.32 + 6.24i)10-s + (−2.79 + 15.8i)11-s + (9.89 + 6.79i)12-s + (−3.24 + 8.91i)13-s + (14.6 + 8.31i)14-s + (7.10 − 7.02i)15-s + (11.9 − 10.6i)16-s + (−26.1 + 15.1i)17-s + ⋯
L(s)  = 1  + (−0.983 + 0.180i)2-s + (0.568 + 0.822i)3-s + (0.934 − 0.355i)4-s + (−0.115 − 0.655i)5-s + (−0.708 − 0.705i)6-s + (−0.921 − 0.772i)7-s + (−0.855 + 0.518i)8-s + (−0.352 + 0.935i)9-s + (0.232 + 0.624i)10-s + (−0.253 + 1.43i)11-s + (0.824 + 0.566i)12-s + (−0.249 + 0.685i)13-s + (1.04 + 0.593i)14-s + (0.473 − 0.468i)15-s + (0.747 − 0.664i)16-s + (−1.54 + 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.470i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.882 - 0.470i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.882 - 0.470i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.137549 + 0.550452i\)
\(L(\frac12)\) \(\approx\) \(0.137549 + 0.550452i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 - 0.361i)T \)
3 \( 1 + (-1.70 - 2.46i)T \)
good5 \( 1 + (0.578 + 3.27i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (6.44 + 5.41i)T + (8.50 + 48.2i)T^{2} \)
11 \( 1 + (2.79 - 15.8i)T + (-113. - 41.3i)T^{2} \)
13 \( 1 + (3.24 - 8.91i)T + (-129. - 108. i)T^{2} \)
17 \( 1 + (26.1 - 15.1i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.88 - 2.24i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-19.4 - 23.2i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (35.3 - 12.8i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (6.91 - 5.80i)T + (166. - 946. i)T^{2} \)
37 \( 1 + (5.81 - 3.35i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-14.6 + 40.2i)T + (-1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-6.02 - 1.06i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-45.4 + 54.1i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 82.0T + 2.80e3T^{2} \)
59 \( 1 + (-1.60 - 9.11i)T + (-3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-5.67 + 6.76i)T + (-646. - 3.66e3i)T^{2} \)
67 \( 1 + (-25.2 + 69.4i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-50.2 + 28.9i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (65.5 - 113. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-13.5 + 4.94i)T + (4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (-23.4 + 8.54i)T + (5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (-138. - 80.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (6.18 - 35.0i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−12.50289351273085442103974481202, −11.08164372022611348272267573535, −10.30338818980803806350306290661, −9.372600621601707990658980560641, −8.940228114281432376219952484799, −7.57986601014284899688060347192, −6.77339963995837702700605482520, −5.04190746846311059129552613089, −3.79742034670476601794971410320, −2.03955441514837497767301062128, 0.37192670686501939888151593596, 2.62273887133875146457042136964, 3.10541623755173473274959023253, 5.99070540082282749415228858084, 6.75273343596684422440470164365, 7.74861784163737921066158895164, 8.849145078747181041912083643224, 9.344725262106630058957026958655, 10.82857095785835123996048612680, 11.45487823028140580836351605217

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