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Properties

Label 2-6e3-72.43-c2-0-15
Degree $2$
Conductor $216$
Sign $0.619 + 0.784i$
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  + (1.75 − 0.965i)2-s + (2.13 − 3.38i)4-s + (−1.50 + 0.871i)5-s + (7.93 + 4.58i)7-s + (0.470 − 7.98i)8-s + (−1.80 + 2.98i)10-s + (4.09 − 7.08i)11-s + (19.9 − 11.5i)13-s + (18.3 + 0.359i)14-s + (−6.88 − 14.4i)16-s − 18.4·17-s − 7.06·19-s + (−0.273 + 6.96i)20-s + (0.320 − 16.3i)22-s + (−9.33 + 5.39i)23-s + ⋯
L(s)  = 1  + (0.875 − 0.482i)2-s + (0.533 − 0.845i)4-s + (−0.301 + 0.174i)5-s + (1.13 + 0.654i)7-s + (0.0587 − 0.998i)8-s + (−0.180 + 0.298i)10-s + (0.371 − 0.644i)11-s + (1.53 − 0.885i)13-s + (1.30 + 0.0256i)14-s + (−0.430 − 0.902i)16-s − 1.08·17-s − 0.371·19-s + (−0.0136 + 0.348i)20-s + (0.0145 − 0.743i)22-s + (−0.405 + 0.234i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.619 + 0.784i$
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: \((2,\ 216,\ (\ :1),\ 0.619 + 0.784i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.44398 - 1.18453i\)
\(L(\frac12)\) \(\approx\) \(2.44398 - 1.18453i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.75 + 0.965i)T \)
3 \( 1 \)
good5 \( 1 + (1.50 - 0.871i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-7.93 - 4.58i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-4.09 + 7.08i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-19.9 + 11.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 18.4T + 289T^{2} \)
19 \( 1 + 7.06T + 361T^{2} \)
23 \( 1 + (9.33 - 5.39i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-14.3 - 8.28i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (1.18 - 0.684i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 59.1iT - 1.36e3T^{2} \)
41 \( 1 + (-19.6 - 34.0i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-8.49 + 14.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (25.4 + 14.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 49.1iT - 2.80e3T^{2} \)
59 \( 1 + (32.8 + 56.8i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-11.8 - 6.86i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (24.8 + 43.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 136. iT - 5.04e3T^{2} \)
73 \( 1 + 120.T + 5.32e3T^{2} \)
79 \( 1 + (82.5 + 47.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-39.2 + 67.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 72.9T + 7.92e3T^{2} \)
97 \( 1 + (25.9 - 44.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.69400206294451009929455847979, −11.29855253828451794675171336388, −10.47400523966154712122881378542, −8.920388241121086641074519802291, −8.038995216391984102315655710049, −6.45595531082367482693400144331, −5.56671432279758997466860445509, −4.34879321302452447950427686629, −3.11950669111078941615302102524, −1.49545669528628721319901152030, 1.91572256536944120745432993239, 4.08895675345336258626866450700, 4.44693685059808878090137385805, 6.05233179451224663130956190142, 7.03525586525408766600275305316, 8.087303930652423731342070967762, 8.913814026115520303423944413161, 10.74683286740578792521599349067, 11.37848711679296348892635873792, 12.26542763510660883557869818262

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