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Properties

Label 2-147-49.43-c1-0-0
Degree $2$
Conductor $147$
Sign $-0.963 + 0.269i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.385 + 0.483i)2-s + (−0.900 − 0.433i)3-s + (0.359 + 1.57i)4-s + (−3.63 − 1.75i)5-s + (0.557 − 0.268i)6-s + (−2.64 − 0.0631i)7-s + (−2.01 − 0.970i)8-s + (0.623 + 0.781i)9-s + (2.25 − 1.08i)10-s + (0.0332 − 0.0416i)11-s + (0.359 − 1.57i)12-s + (−0.237 + 0.297i)13-s + (1.05 − 1.25i)14-s + (2.51 + 3.15i)15-s + (−1.66 + 0.803i)16-s + (−0.172 + 0.755i)17-s + ⋯
L(s)  = 1  + (−0.272 + 0.341i)2-s + (−0.520 − 0.250i)3-s + (0.179 + 0.788i)4-s + (−1.62 − 0.783i)5-s + (0.227 − 0.109i)6-s + (−0.999 − 0.0238i)7-s + (−0.712 − 0.343i)8-s + (0.207 + 0.260i)9-s + (0.711 − 0.342i)10-s + (0.0100 − 0.0125i)11-s + (0.103 − 0.455i)12-s + (−0.0657 + 0.0825i)13-s + (0.280 − 0.335i)14-s + (0.650 + 0.815i)15-s + (−0.416 + 0.200i)16-s + (−0.0418 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.963 + 0.269i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.963 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00478077 - 0.0348290i\)
\(L(\frac12)\) \(\approx\) \(0.00478077 - 0.0348290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (2.64 + 0.0631i)T \)
good2 \( 1 + (0.385 - 0.483i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (3.63 + 1.75i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-0.0332 + 0.0416i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.237 - 0.297i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (0.172 - 0.755i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + (0.445 + 1.95i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (1.94 - 8.52i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 5.67T + 31T^{2} \)
37 \( 1 + (-2.39 + 10.5i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (1.87 + 0.902i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (6.37 - 3.06i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (6.67 - 8.37i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (1.51 + 6.65i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (6.38 - 3.07i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-1.24 + 5.45i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + (2.39 + 10.5i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.54 + 1.93i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + (-0.559 - 0.702i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-7.55 - 9.47i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 9.99T + 97T^{2} \)
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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

−12.99927066883395945131586360864, −12.57651435038738373367446065258, −11.84489140994858211297577614938, −10.82106844178819106072879319767, −9.141589152664246480994488561109, −8.262694380515544339137724563540, −7.34046134159106854776888319901, −6.38629443478250450210435758141, −4.52876162807196819845237324678, −3.38985766513405038947173573291, 0.03839408850449409308584889554, 3.01675092712218267751980933784, 4.39590117736919082511292195609, 6.13719058581789918771036394774, 6.95752127270433417992373200492, 8.361273084612131093530492511355, 9.835682160704685149778852139255, 10.49018717554528285782628362929, 11.55991354097299607223702857926, 11.95028763922648397872729017057

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