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Properties

Label 2-147-21.5-c1-0-8
Degree $2$
Conductor $147$
Sign $-0.930 + 0.365i$
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  + (−1.09 − 0.629i)2-s + (−0.353 − 1.69i)3-s + (−0.207 − 0.358i)4-s + (1.16 − 2.01i)5-s + (−0.681 + 2.07i)6-s + 3.04i·8-s + (−2.74 + 1.20i)9-s + (−2.53 + 1.46i)10-s + (−1.54 + 0.890i)11-s + (−0.534 + 0.478i)12-s − 4.46i·13-s + (−3.82 − 1.25i)15-s + (1.5 − 2.59i)16-s + (0.481 + 0.834i)17-s + (3.75 + 0.422i)18-s + (−1.87 − 1.08i)19-s + ⋯
L(s)  = 1  + (−0.771 − 0.445i)2-s + (−0.204 − 0.978i)3-s + (−0.103 − 0.179i)4-s + (0.520 − 0.901i)5-s + (−0.278 + 0.845i)6-s + 1.07i·8-s + (−0.916 + 0.400i)9-s + (−0.802 + 0.463i)10-s + (−0.465 + 0.268i)11-s + (−0.154 + 0.138i)12-s − 1.23i·13-s + (−0.988 − 0.325i)15-s + (0.375 − 0.649i)16-s + (0.116 + 0.202i)17-s + (0.884 + 0.0994i)18-s + (−0.430 − 0.248i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.930 + 0.365i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.930 + 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117269 - 0.619333i\)
\(L(\frac12)\) \(\approx\) \(0.117269 - 0.619333i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.353 + 1.69i)T \)
7 \( 1 \)
good2 \( 1 + (1.09 + 0.629i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.16 + 2.01i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.54 - 0.890i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.46iT - 13T^{2} \)
17 \( 1 + (-0.481 - 0.834i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.87 + 1.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.72 - 2.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.86iT - 29T^{2} \)
31 \( 1 + (-0.937 + 0.541i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.53 + 4.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.25T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-0.282 + 0.488i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.26 + 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.65 + 8.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.51 + 3.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.24 - 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 + (-5.57 + 3.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.41 - 4.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (-4.17 + 7.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−12.79648093350308225591527669063, −11.51619062402875139261443945196, −10.54691069822423091124590055758, −9.486851668130559527422602912862, −8.489588403056639976638598355312, −7.64668472566276269003374363915, −5.91405644379200580150710150284, −5.10096314219290001512994648866, −2.36428661541490663536266079636, −0.855945004991916102711359088139, 3.05695956599226385659574534741, 4.51651198655836770544804710348, 6.15351085359012871115501946947, 7.11846143603744269454800714854, 8.577771773251788736949800238892, 9.355791439154121632652076987549, 10.31433226927537368332956810160, 11.05591406095963276308061424110, 12.38975141146218753442731870390, 13.77376284463174941298751139274

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