Properties

Label 2-6e3-72.29-c2-0-18
Degree $2$
Conductor $216$
Sign $0.845 + 0.534i$
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  + (1.87 + 0.698i)2-s + (3.02 + 2.61i)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 + (17.4 − 14.3i)14-s + (2.29 + 15.8i)16-s + 4.29i·17-s + 11.2i·19-s + (6.01 − 31.3i)20-s + (3.27 − 2.70i)22-s + (3.40 − 1.96i)23-s + ⋯
L(s)  = 1  + (0.937 + 0.349i)2-s + (0.756 + 0.654i)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 + (1.24 − 1.02i)14-s + (0.143 + 0.989i)16-s + 0.252i·17-s + 0.590i·19-s + (0.300 − 1.56i)20-s + (0.148 − 0.122i)22-s + (0.147 − 0.0853i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.845 + 0.534i$
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: \((2,\ 216,\ (\ :1),\ 0.845 + 0.534i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.45292 - 0.710449i\)
\(L(\frac12)\) \(\approx\) \(2.45292 - 0.710449i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 - 0.698i)T \)
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

−12.10198433858408610451322472058, −11.37065452857346386115313330368, −10.34562332076592620629769353204, −8.500492990409268694438466176720, −8.006127308963029288202799292596, −6.95262795727746709217575049438, −5.40876864014329861946575206415, −4.44283946676596041645975727608, −3.73024479047005721528514115507, −1.22774863822418062801069587979, 2.21009931984095158207787159682, 3.27179619862997977005094778253, 4.62103356623736040405175380349, 5.88461893229252813640243381944, 6.88533457304483927536776452870, 7.970194360813244721088028041388, 9.378700668737236042063724618967, 10.78307174270815345261961756195, 11.39557126312175415074124243388, 11.91448440193326462071566136742

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