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Properties

Label 2-6e3-72.43-c2-0-2
Degree $2$
Conductor $216$
Sign $-0.966 + 0.256i$
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  + (−1.67 + 1.08i)2-s + (1.62 − 3.65i)4-s + (−5.42 + 3.12i)5-s + (5.96 + 3.44i)7-s + (1.24 + 7.90i)8-s + (5.68 − 11.1i)10-s + (5.38 − 9.32i)11-s + (−15.3 + 8.88i)13-s + (−13.7 + 0.719i)14-s + (−10.7 − 11.8i)16-s + 0.681·17-s − 26.6·19-s + (2.61 + 24.8i)20-s + (1.12 + 21.5i)22-s + (−22.8 + 13.1i)23-s + ⋯
L(s)  = 1  + (−0.838 + 0.544i)2-s + (0.406 − 0.913i)4-s + (−1.08 + 0.625i)5-s + (0.851 + 0.491i)7-s + (0.156 + 0.987i)8-s + (0.568 − 1.11i)10-s + (0.489 − 0.847i)11-s + (−1.18 + 0.683i)13-s + (−0.982 + 0.0513i)14-s + (−0.668 − 0.743i)16-s + 0.0400·17-s − 1.40·19-s + (0.130 + 1.24i)20-s + (0.0511 + 0.977i)22-s + (−0.992 + 0.572i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.966 + 0.256i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0332229 - 0.254205i\)
\(L(\frac12)\) \(\approx\) \(0.0332229 - 0.254205i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.67 - 1.08i)T \)
3 \( 1 \)
good5 \( 1 + (5.42 - 3.12i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-5.96 - 3.44i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-5.38 + 9.32i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (15.3 - 8.88i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 0.681T + 289T^{2} \)
19 \( 1 + 26.6T + 361T^{2} \)
23 \( 1 + (22.8 - 13.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (26.8 + 15.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (19.9 - 11.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 33.4iT - 1.36e3T^{2} \)
41 \( 1 + (13.4 + 23.3i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (16.5 - 28.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-10.6 - 6.13i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 49.2iT - 2.80e3T^{2} \)
59 \( 1 + (-19.0 - 32.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-65.7 - 37.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (27.4 + 47.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 35.6iT - 5.04e3T^{2} \)
73 \( 1 - 113.T + 5.32e3T^{2} \)
79 \( 1 + (-30.8 - 17.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (58.7 - 101. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 58.3T + 7.92e3T^{2} \)
97 \( 1 + (63.7 - 110. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.16905153260790922580992014499, −11.41641890673694621510202971139, −10.81347402882890346425612318506, −9.488821421537476283839737511390, −8.486343831741284027000419584973, −7.71170876143499534558237215888, −6.80217789914008751127532352241, −5.55516778829040538231277381764, −4.08218632131782357845486148408, −2.12719972987115075619035472466, 0.17820595404622332024556293894, 1.95990985061198551004392518319, 3.88964579783024666794798905277, 4.75290217229310981960079084712, 6.92450572338418419921051153618, 7.86032293994865655482057203245, 8.415018912956823128768769643810, 9.678134637414233476647410099599, 10.59968082377389580695397856385, 11.57936103686626960424558281344

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