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Properties

Label 2-6e3-72.61-c1-0-7
Degree $2$
Conductor $216$
Sign $0.906 + 0.422i$
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  + (1.36 − 0.366i)2-s + (1.73 − i)4-s + (1.73 − i)5-s + (−2 + 3.46i)7-s + (1.99 − 2i)8-s + (1.99 − 2i)10-s + (−2.59 − 1.5i)11-s + (1.73 − i)13-s + (−1.46 + 5.46i)14-s + (1.99 − 3.46i)16-s − 5·17-s + i·19-s + (1.99 − 3.46i)20-s + (−4.09 − 1.09i)22-s + (1 + 1.73i)23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (0.774 − 0.447i)5-s + (−0.755 + 1.30i)7-s + (0.707 − 0.707i)8-s + (0.632 − 0.632i)10-s + (−0.783 − 0.452i)11-s + (0.480 − 0.277i)13-s + (−0.391 + 1.46i)14-s + (0.499 − 0.866i)16-s − 1.21·17-s + 0.229i·19-s + (0.447 − 0.774i)20-s + (−0.873 − 0.234i)22-s + (0.208 + 0.361i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.906 + 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07033 - 0.458981i\)
\(L(\frac12)\) \(\approx\) \(2.07033 - 0.458981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 \)
good5 \( 1 + (-1.73 + i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.52 + 5.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.3 - 6i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.48335729396583527390590189808, −11.50284980469511585215836280874, −10.44085417466216955794832872097, −9.425844864693216776246554208264, −8.431662777003552260151574237361, −6.68144037078655771021164295697, −5.78243200103484308714693569016, −5.08061973141520009686416015880, −3.29884724948060516704191743938, −2.11554593785227139652540245546, 2.36332118847499301302913558219, 3.76294563482214022757096069319, 4.92878379485521351559472110760, 6.44053010502898545213009758947, 6.82722978419181667164956850862, 8.107254999974141425733612226723, 9.770116090951147465538491338719, 10.55889326384078634580707331686, 11.39181425957411070834864329612, 12.88463597714765421131100185509

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