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Properties

Label 2-6e3-72.29-c2-0-10
Degree $2$
Conductor $216$
Sign $-0.00538 - 0.999i$
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  + (1.54 + 1.27i)2-s + (0.754 + 3.92i)4-s + (3.98 + 6.90i)5-s + (5.64 − 9.78i)7-s + (−3.84 + 7.01i)8-s + (−2.65 + 15.7i)10-s + (−1.06 + 1.83i)11-s + (−6.03 + 3.48i)13-s + (21.1 − 7.88i)14-s + (−14.8 + 5.92i)16-s + 4.29i·17-s − 11.2i·19-s + (−24.1 + 20.8i)20-s + (−3.97 + 1.48i)22-s + (3.40 − 1.96i)23-s + ⋯
L(s)  = 1  + (0.770 + 0.636i)2-s + (0.188 + 0.982i)4-s + (0.797 + 1.38i)5-s + (0.806 − 1.39i)7-s + (−0.480 + 0.877i)8-s + (−0.265 + 1.57i)10-s + (−0.0964 + 0.166i)11-s + (−0.464 + 0.268i)13-s + (1.51 − 0.563i)14-s + (−0.928 + 0.370i)16-s + 0.252i·17-s − 0.590i·19-s + (−1.20 + 1.04i)20-s + (−0.180 + 0.0673i)22-s + (0.147 − 0.0853i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00538 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00538 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.00538 - 0.999i$
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: \((2,\ 216,\ (\ :1),\ -0.00538 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.88417 + 1.89433i\)
\(L(\frac12)\) \(\approx\) \(1.88417 + 1.89433i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.54 - 1.27i)T \)
3 \( 1 \)
good5 \( 1 + (-3.98 - 6.90i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-5.64 + 9.78i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (1.06 - 1.83i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (6.03 - 3.48i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 4.29iT - 289T^{2} \)
19 \( 1 + 11.2iT - 361T^{2} \)
23 \( 1 + (-3.40 + 1.96i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (0.512 - 0.887i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-4.18 - 7.25i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 58.9iT - 1.36e3T^{2} \)
41 \( 1 + (-48.6 + 28.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-49.8 - 28.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (44.8 + 25.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 42.6T + 2.80e3T^{2} \)
59 \( 1 + (25.6 + 44.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (42.9 + 24.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-85.1 + 49.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 31.4iT - 5.04e3T^{2} \)
73 \( 1 - 85.4T + 5.32e3T^{2} \)
79 \( 1 + (6.61 - 11.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-41.8 + 72.5i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 105. iT - 7.92e3T^{2} \)
97 \( 1 + (9.54 - 16.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.60036422297726352310305572418, −11.11419152389553936834916803289, −10.76734245193531380097319528849, −9.464549868065735297773657303143, −7.80729343572108114708909870537, −7.13750468864304141328960293883, −6.31517835128454901756075475656, −4.95993518175322410355866132723, −3.76541233651471342222898299006, −2.32331102559701852432885946617, 1.40688541076392688785694924782, 2.59911821569712810114630814840, 4.60588756826267265489597565649, 5.33470431235657938377392989960, 6.06763334662514543588502340621, 8.113770416614340502037834162866, 9.121809220440621116120329124820, 9.814403356065262148683164669631, 11.17720023086402330548797911867, 12.14269577014005771519776703876

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