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Properties

Label 2-147-7.4-c5-0-8
Degree $2$
Conductor $147$
Sign $0.0725 + 0.997i$
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  + (−4.10 − 7.10i)2-s + (−4.5 + 7.79i)3-s + (−17.6 + 30.6i)4-s + (−14.6 − 25.3i)5-s + 73.8·6-s + 27.7·8-s + (−40.5 − 70.1i)9-s + (−119. + 207. i)10-s + (−188. + 326. i)11-s + (−159. − 275. i)12-s + 509.·13-s + 263.·15-s + (452. + 783. i)16-s + (−802. + 1.39e3i)17-s + (−332. + 575. i)18-s + (−1.07e3 − 1.86e3i)19-s + ⋯
L(s)  = 1  + (−0.725 − 1.25i)2-s + (−0.288 + 0.499i)3-s + (−0.552 + 0.957i)4-s + (−0.261 − 0.452i)5-s + 0.837·6-s + 0.153·8-s + (−0.166 − 0.288i)9-s + (−0.379 + 0.657i)10-s + (−0.470 + 0.814i)11-s + (−0.319 − 0.552i)12-s + 0.836·13-s + 0.301·15-s + (0.441 + 0.764i)16-s + (−0.673 + 1.16i)17-s + (−0.241 + 0.418i)18-s + (−0.685 − 1.18i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8597057629\)
\(L(\frac12)\) \(\approx\) \(0.8597057629\)
\(L(\frac{7}{2})\) 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 \( 1 + (4.10 + 7.10i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (14.6 + 25.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (188. - 326. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 509.T + 3.71e5T^{2} \)
17 \( 1 + (802. - 1.39e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.07e3 + 1.86e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.21e3 - 3.84e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.77e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.57e3 + 6.19e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.78e3 + 4.82e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 9.91e3T + 1.15e8T^{2} \)
43 \( 1 + 4.88e3T + 1.47e8T^{2} \)
47 \( 1 + (4.63e3 + 8.02e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-4.62e3 + 8.00e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-7.06e3 + 1.22e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-9.01e3 - 1.56e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.32e4 + 4.02e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.42e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.04e4 + 5.27e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.57e4 - 6.18e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.15e4T + 3.93e9T^{2} \)
89 \( 1 + (3.91e4 + 6.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.51e5T + 8.58e9T^{2} \)
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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

−11.58322781690039135467838316792, −10.86590785171052728200747469509, −10.04482204968888367988345029136, −9.019400622042319330119384183452, −8.250938178089046788364976937750, −6.47701130014100605914325521206, −4.84981232753615582588047627124, −3.64786187903663186704431163738, −2.11316279803754402268277645691, −0.62610657226854368535548144900, 0.75247912593173601622975537789, 3.00076554229018547765915176319, 5.09075672067411186379103671752, 6.40015504109590319696457080618, 6.89586639485439963404188483161, 8.225222745872243650081194479335, 8.697325635188755279447497199518, 10.28232898905529654380140133676, 11.21914005913628593776357777607, 12.39817016108677389637744834931

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