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Properties

Label 2-6e3-9.7-c7-0-16
Degree $2$
Conductor $216$
Sign $-0.865 + 0.500i$
Analytic cond. $67.4751$
Root an. cond. $8.21432$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−130. − 225. i)5-s + (−113. + 197. i)7-s + (1.73e3 − 3.01e3i)11-s + (728. + 1.26e3i)13-s + 1.46e4·17-s + 6.09e3·19-s + (1.63e3 + 2.83e3i)23-s + (5.03e3 − 8.72e3i)25-s + (5.51e4 − 9.54e4i)29-s + (1.69e4 + 2.93e4i)31-s + 5.94e4·35-s − 1.05e5·37-s + (−3.37e5 − 5.84e5i)41-s + (−3.21e5 + 5.56e5i)43-s + (1.22e5 − 2.11e5i)47-s + ⋯
L(s)  = 1  + (−0.466 − 0.808i)5-s + (−0.125 + 0.217i)7-s + (0.393 − 0.682i)11-s + (0.0919 + 0.159i)13-s + 0.724·17-s + 0.203·19-s + (0.0280 + 0.0485i)23-s + (0.0644 − 0.111i)25-s + (0.419 − 0.726i)29-s + (0.102 + 0.176i)31-s + 0.234·35-s − 0.342·37-s + (−0.764 − 1.32i)41-s + (−0.615 + 1.06i)43-s + (0.171 − 0.297i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.865 + 0.500i$
Analytic conductor: \(67.4751\)
Root analytic conductor: \(8.21432\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :7/2),\ -0.865 + 0.500i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.025335597\)
\(L(\frac12)\) \(\approx\) \(1.025335597\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (130. + 225. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (113. - 197. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-1.73e3 + 3.01e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-728. - 1.26e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 - 1.46e4T + 4.10e8T^{2} \)
19 \( 1 - 6.09e3T + 8.93e8T^{2} \)
23 \( 1 + (-1.63e3 - 2.83e3i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-5.51e4 + 9.54e4i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (-1.69e4 - 2.93e4i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + 1.05e5T + 9.49e10T^{2} \)
41 \( 1 + (3.37e5 + 5.84e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (3.21e5 - 5.56e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (-1.22e5 + 2.11e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + 1.83e6T + 1.17e12T^{2} \)
59 \( 1 + (-1.06e6 - 1.84e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-8.33e5 + 1.44e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (2.39e6 + 4.15e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + 2.50e6T + 9.09e12T^{2} \)
73 \( 1 + 4.46e4T + 1.10e13T^{2} \)
79 \( 1 + (-3.33e6 + 5.77e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (9.91e5 - 1.71e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + 8.71e6T + 4.42e13T^{2} \)
97 \( 1 + (-4.29e5 + 7.44e5i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−10.69731322534273696465556422017, −9.520302143335648515521745033884, −8.633743967060373401031317892187, −7.83042347130757522560325641651, −6.48529184922778840467496573849, −5.38400871893626306658200534746, −4.25373700443113264491010424709, −3.08656095370814975150970746646, −1.40005897460826489217233407013, −0.26630219250633975951848522493, 1.34435702116533199103366153385, 2.90495590283040527234699307697, 3.87742252929136504985028528062, 5.20593937442276549249154609249, 6.61707418708070375187863799846, 7.30970627436165783366249985392, 8.389987912469316861127829209062, 9.681324904040455304997053494957, 10.46331453239413502943465762822, 11.45679022354250542651806888187

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