Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.853 - 0.520i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.18 + 2.67i)2-s + (3.68 + 3.66i)3-s + (1.62 − 9.19i)4-s + (−14.0 + 5.09i)5-s + (−21.5 − 1.84i)6-s + (0.758 + 4.30i)7-s + (2.78 + 4.82i)8-s + (0.0900 + 26.9i)9-s + (31.0 − 53.7i)10-s + (60.8 + 22.1i)11-s + (39.7 − 27.8i)12-s + (−2.75 − 2.31i)13-s + (−13.9 − 11.6i)14-s + (−70.2 − 32.6i)15-s + (48.3 + 17.6i)16-s + (−18.0 + 31.2i)17-s + ⋯
L(s)  = 1  + (−1.12 + 0.946i)2-s + (0.708 + 0.705i)3-s + (0.202 − 1.14i)4-s + (−1.25 + 0.455i)5-s + (−1.46 − 0.125i)6-s + (0.0409 + 0.232i)7-s + (0.123 + 0.213i)8-s + (0.00333 + 0.999i)9-s + (0.981 − 1.69i)10-s + (1.66 + 0.606i)11-s + (0.955 − 0.671i)12-s + (−0.0588 − 0.0494i)13-s + (−0.265 − 0.223i)14-s + (−1.20 − 0.561i)15-s + (0.756 + 0.275i)16-s + (−0.257 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.853 - 0.520i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -0.853 - 0.520i)$
$L(2)$  $\approx$  $0.187181 + 0.666367i$
$L(\frac12)$  $\approx$  $0.187181 + 0.666367i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-3.68 - 3.66i)T \)
good2 \( 1 + (3.18 - 2.67i)T + (1.38 - 7.87i)T^{2} \)
5 \( 1 + (14.0 - 5.09i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-0.758 - 4.30i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (-60.8 - 22.1i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (2.75 + 2.31i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (18.0 - 31.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-37.4 - 64.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-36.4 + 206. i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-68.7 + 57.6i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (4.29 - 24.3i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-38.9 + 67.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (91.8 + 77.0i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (294. + 107. i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-50.7 - 287. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 - 512.T + 1.48e5T^{2} \)
59 \( 1 + (-3.32 + 1.20i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (62.7 + 355. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-144. - 121. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (243. - 422. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (24.5 + 42.5i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-442. + 371. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-407. + 342. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-358. - 621. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-312. - 113. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.05850433129688838286625960927, −16.14334750255929074277738801846, −15.08081742426770979375961603602, −14.55221132896325079295611732657, −12.06015069543545788475266682880, −10.40764841200409322134021705701, −9.065961563659183399908032239306, −8.092415554890415484502509168696, −6.80717266208482004472767716230, −3.95576356354497377645418500407, 1.02285008517443497630368547100, 3.51765351858982572606721730752, 7.25339107669139340513411544180, 8.535288267809056414740217088511, 9.370620371743687805691146974142, 11.50857063512380356607011469136, 11.92411420164395834783089405171, 13.64594872298825718124925449199, 15.15674682102620023432983261260, 16.74938677632164966661713341803

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