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Properties

Label 2-6e3-72.29-c2-0-11
Degree $2$
Conductor $216$
Sign $0.808 + 0.589i$
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  + (−1.45 + 1.37i)2-s + (0.206 − 3.99i)4-s + (−1.89 − 3.28i)5-s + (−5.70 + 9.88i)7-s + (5.20 + 6.07i)8-s + (7.27 + 2.15i)10-s + (3.47 − 6.01i)11-s + (14.6 − 8.48i)13-s + (−5.33 − 22.1i)14-s + (−15.9 − 1.64i)16-s − 22.9i·17-s − 21.7i·19-s + (−13.5 + 6.89i)20-s + (3.24 + 13.4i)22-s + (13.7 − 7.93i)23-s + ⋯
L(s)  = 1  + (−0.725 + 0.688i)2-s + (0.0515 − 0.998i)4-s + (−0.379 − 0.656i)5-s + (−0.815 + 1.41i)7-s + (0.650 + 0.759i)8-s + (0.727 + 0.215i)10-s + (0.315 − 0.546i)11-s + (1.13 − 0.652i)13-s + (−0.381 − 1.58i)14-s + (−0.994 − 0.102i)16-s − 1.35i·17-s − 1.14i·19-s + (−0.675 + 0.344i)20-s + (0.147 + 0.613i)22-s + (0.597 − 0.344i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.589i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.808 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.808 + 0.589i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.808 + 0.589i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.772263 - 0.251591i\)
\(L(\frac12)\) \(\approx\) \(0.772263 - 0.251591i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.45 - 1.37i)T \)
3 \( 1 \)
good5 \( 1 + (1.89 + 3.28i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (5.70 - 9.88i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.47 + 6.01i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-14.6 + 8.48i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 22.9iT - 289T^{2} \)
19 \( 1 + 21.7iT - 361T^{2} \)
23 \( 1 + (-13.7 + 7.93i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-6.57 + 11.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-3.45 - 5.98i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 1.75iT - 1.36e3T^{2} \)
41 \( 1 + (33.1 - 19.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (10.9 + 6.30i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28.8 - 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 1.96T + 2.80e3T^{2} \)
59 \( 1 + (-10.5 - 18.2i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (48.0 + 27.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-75.4 + 43.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 38.7iT - 5.04e3T^{2} \)
73 \( 1 - 31.7T + 5.32e3T^{2} \)
79 \( 1 + (68.7 - 119. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-33.5 + 58.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 159. iT - 7.92e3T^{2} \)
97 \( 1 + (-42.5 + 73.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.87453922663363656588842314370, −11.00421344783242180896857104259, −9.623452279477683665720447378376, −8.851857603291700639378036837518, −8.353336485272802094959546958917, −6.85042141952179244897307901267, −5.90994624520595516687737165259, −4.91587566164240780557244602118, −2.86274558356783044367370240518, −0.63556473195114284264441676879, 1.41920803138544051186996871035, 3.49715342504636123581189160692, 3.98959170120946444462586722768, 6.49102700114401686628345206020, 7.19165211566896391748762176505, 8.268632421805652098002918040749, 9.466198869614016950566286117466, 10.46443934374673992325039438222, 10.86356860592847153092539039950, 12.02346290531185617764928140779

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