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Properties

Label 2-147-7.4-c3-0-7
Degree $2$
Conductor $147$
Sign $-0.991 + 0.126i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.65 − 4.59i)2-s + (−1.5 + 2.59i)3-s + (−10.0 + 17.4i)4-s + (2.78 + 4.81i)5-s + 15.9·6-s + 64.6·8-s + (−4.5 − 7.79i)9-s + (14.7 − 25.5i)10-s + (6.95 − 12.0i)11-s + (−30.2 − 52.4i)12-s − 38.6·13-s − 16.6·15-s + (−90.8 − 157. i)16-s + (21.7 − 37.6i)17-s + (−23.8 + 41.3i)18-s + (−54.5 − 94.4i)19-s + ⋯
L(s)  = 1  + (−0.938 − 1.62i)2-s + (−0.288 + 0.499i)3-s + (−1.26 + 2.18i)4-s + (0.248 + 0.430i)5-s + 1.08·6-s + 2.85·8-s + (−0.166 − 0.288i)9-s + (0.466 − 0.808i)10-s + (0.190 − 0.330i)11-s + (−0.728 − 1.26i)12-s − 0.825·13-s − 0.287·15-s + (−1.41 − 2.45i)16-s + (0.310 − 0.537i)17-s + (−0.312 + 0.541i)18-s + (−0.658 − 1.14i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0304974 - 0.480576i\)
\(L(\frac12)\) \(\approx\) \(0.0304974 - 0.480576i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (2.65 + 4.59i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-2.78 - 4.81i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-6.95 + 12.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 38.6T + 2.19e3T^{2} \)
17 \( 1 + (-21.7 + 37.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (54.5 + 94.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-37.4 - 64.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 72.3T + 2.43e4T^{2} \)
31 \( 1 + (-32.0 + 55.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (94.3 + 163. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 24.7T + 6.89e4T^{2} \)
43 \( 1 + 243.T + 7.95e4T^{2} \)
47 \( 1 + (310. + 537. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-143. + 249. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-262. + 454. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (191. + 332. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (99.0 - 171. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 785.T + 3.57e5T^{2} \)
73 \( 1 + (165. - 286. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (218. + 379. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 241.T + 5.71e5T^{2} \)
89 \( 1 + (-792. - 1.37e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 79.2T + 9.12e5T^{2} \)
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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

−11.68201185187394458494836663489, −11.04803891567492805974257801572, −10.10960073659257443711679379948, −9.426545073787593846364323540579, −8.416578271116148839773269877707, −6.98390669296325094685326120138, −4.93582705870822397812164409094, −3.50680614255033585638659901208, −2.30210140327570921109045401688, −0.35351063831337415571864782813, 1.41367341242992595578418365154, 4.75163717551516524574216978547, 5.81502548410588056292781192190, 6.76870367834723335332825582149, 7.75077740815935107501792162544, 8.636064037606274275680432654322, 9.678975987518505234517770928003, 10.58731328731501758959777918282, 12.26480145027554144360344121557, 13.26920964917700338292765283254

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