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Properties

Label 2-147-7.2-c5-0-30
Degree $2$
Conductor $147$
Sign $-0.991 - 0.126i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.59 − 7.95i)2-s + (4.5 + 7.79i)3-s + (−26.1 − 45.2i)4-s + (11.0 − 19.1i)5-s + 82.6·6-s − 186.·8-s + (−40.5 + 70.1i)9-s + (−101. − 175. i)10-s + (−208. − 360. i)11-s + (235. − 407. i)12-s − 797.·13-s + 198.·15-s + (−18.6 + 32.3i)16-s + (−687. − 1.19e3i)17-s + (371. + 644. i)18-s + (1.15e3 − 2.00e3i)19-s + ⋯
L(s)  = 1  + (0.811 − 1.40i)2-s + (0.288 + 0.499i)3-s + (−0.817 − 1.41i)4-s + (0.197 − 0.341i)5-s + 0.937·6-s − 1.02·8-s + (−0.166 + 0.288i)9-s + (−0.320 − 0.554i)10-s + (−0.519 − 0.899i)11-s + (0.471 − 0.817i)12-s − 1.30·13-s + 0.227·15-s + (−0.0182 + 0.0315i)16-s + (−0.577 − 0.999i)17-s + (0.270 + 0.468i)18-s + (0.734 − 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.334681374\)
\(L(\frac12)\) \(\approx\) \(2.334681374\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.5 - 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (-4.59 + 7.95i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-11.0 + 19.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (208. + 360. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 797.T + 3.71e5T^{2} \)
17 \( 1 + (687. + 1.19e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.15e3 + 2.00e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-477. + 827. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 7.03e3T + 2.05e7T^{2} \)
31 \( 1 + (-630. - 1.09e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (4.88e3 - 8.46e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 5.40e3T + 1.15e8T^{2} \)
43 \( 1 - 1.96e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.02e3 + 1.78e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (9.01e3 + 1.56e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-3.71e3 - 6.43e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.74e3 + 3.02e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (7.92e3 + 1.37e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 5.81e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.95e4 - 3.38e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.88e3 - 8.45e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 7.03e4T + 3.93e9T^{2} \)
89 \( 1 + (-7.21e4 + 1.24e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 7.93e4T + 8.58e9T^{2} \)
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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

−11.49438990463545628546477636352, −10.91281227628661464338405879748, −9.720271996588236137951028684577, −9.050706312306071607252860432709, −7.36801561970985846652697332323, −5.32577318534173093049250012105, −4.72957975982543698217639054564, −3.23844859942235457790333170546, −2.36397423999022205422348389040, −0.55243391021753936573227649399, 2.17325847057059899031331605466, 3.91480400933301333557298442757, 5.21881478637155298753839855213, 6.24064957355662710594290625112, 7.38598943051586544447580285725, 7.80361885954052505471196119962, 9.314294834162840189750595297530, 10.57305772372926912738942836236, 12.34487474720154039834614486751, 12.79816465498666765827223931725

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