Properties

Degree $2$
Conductor $147$
Sign $0.386 - 0.922i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (0.500 − 0.866i)4-s + (1 + 1.73i)5-s − 0.999·6-s + 3·8-s + (−0.499 − 0.866i)9-s + (−0.999 + 1.73i)10-s + (−2 + 3.46i)11-s + (0.499 + 0.866i)12-s − 2·13-s − 1.99·15-s + (0.500 + 0.866i)16-s + (3 − 5.19i)17-s + (0.499 − 0.866i)18-s + (−2 − 3.46i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s + (0.447 + 0.774i)5-s − 0.408·6-s + 1.06·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (−0.603 + 1.04i)11-s + (0.144 + 0.249i)12-s − 0.554·13-s − 0.516·15-s + (0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + (0.117 − 0.204i)18-s + (−0.458 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Motivic weight: \(1\)
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13573 + 0.755468i\)
\(L(\frac12)\) \(\approx\) \(1.13573 + 0.755468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−13.56775609668087760755850906591, −12.24779454433049900567752070553, −10.99352499726750975058624035967, −10.24219156000314307007322760173, −9.457711587475664148183170594594, −7.50422561668344633323687756596, −6.77557955340642932004863758125, −5.52741282057284022275560102929, −4.63457634943671944158429689057, −2.52632112391922761361217206543, 1.73281414074435367102549418908, 3.40825931087961282193072698863, 5.03255330496125635583567660679, 6.18865728835629433184375431217, 7.73774369923175224258379986051, 8.510065100013018042821434896286, 10.11841087951395740715547439771, 11.03316644612270975077018765656, 12.10556268985430732450527878656, 12.80677120594093965503210946899

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