Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.345 + 0.938i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.553 − 3.14i)2-s + (−0.697 − 5.14i)3-s + (−2.03 − 0.742i)4-s + (−10.9 + 9.21i)5-s + (−16.5 − 0.660i)6-s + (30.2 − 11.0i)7-s + (9.29 − 16.1i)8-s + (−26.0 + 7.18i)9-s + (22.8 + 39.5i)10-s + (15.6 + 13.0i)11-s + (−2.39 + 11.0i)12-s + (7.14 + 40.5i)13-s + (−17.8 − 101. i)14-s + (55.1 + 50.1i)15-s + (−58.7 − 49.2i)16-s + (14.6 + 25.4i)17-s + ⋯
L(s)  = 1  + (0.195 − 1.11i)2-s + (−0.134 − 0.990i)3-s + (−0.254 − 0.0927i)4-s + (−0.982 + 0.824i)5-s + (−1.12 − 0.0449i)6-s + (1.63 − 0.594i)7-s + (0.410 − 0.711i)8-s + (−0.963 + 0.266i)9-s + (0.722 + 1.25i)10-s + (0.427 + 0.359i)11-s + (−0.0577 + 0.265i)12-s + (0.152 + 0.864i)13-s + (−0.340 − 1.93i)14-s + (0.948 + 0.862i)15-s + (−0.917 − 0.769i)16-s + (0.209 + 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.345 + 0.938i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (25, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -0.345 + 0.938i)$
$L(2)$  $\approx$  $0.730903 - 1.04739i$
$L(\frac12)$  $\approx$  $0.730903 - 1.04739i$
$L(\frac{5}{2})$   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 \( 1 + (0.697 + 5.14i)T \)
good2 \( 1 + (-0.553 + 3.14i)T + (-7.51 - 2.73i)T^{2} \)
5 \( 1 + (10.9 - 9.21i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (-30.2 + 11.0i)T + (262. - 220. i)T^{2} \)
11 \( 1 + (-15.6 - 13.0i)T + (231. + 1.31e3i)T^{2} \)
13 \( 1 + (-7.14 - 40.5i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (-14.6 - 25.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (31.1 - 53.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (97.4 + 35.4i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (-0.781 + 4.43i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (47.4 + 17.2i)T + (2.28e4 + 1.91e4i)T^{2} \)
37 \( 1 + (29.6 + 51.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-33.1 - 188. i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (270. + 227. i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (-282. + 102. i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 + 128.T + 1.48e5T^{2} \)
59 \( 1 + (-440. + 369. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (633. - 230. i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (-127. - 720. i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (223. + 386. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-408. + 707. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-13.9 + 78.8i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (177. - 1.00e3i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (241. - 417. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.01e3 + 850. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−16.75769949215748141417270553103, −14.79480888987900712854570762682, −13.86840395903082568395695743232, −12.17568007210695630599672089020, −11.50772974102672236875413891047, −10.68028758789669451683901292084, −8.035337291342313741064453485381, −6.97320259875625741302973948338, −4.04362614121773580430116701209, −1.77130883060899360815301648393, 4.52969449081657630288243718884, 5.54818448747829124047066869878, 7.943682741828944769422426889460, 8.696252068222601451360362274995, 11.02046891440134370821781951331, 11.89239880339352060834313520809, 14.16670010144031270467311311816, 15.18006719430482467752772216043, 15.77334561176772869841834963143, 16.85152606419720470749319197399

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