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Properties

Label 2-147-21.2-c2-0-7
Degree $2$
Conductor $147$
Sign $-0.506 - 0.862i$
Analytic cond. $4.00545$
Root an. cond. $2.00136$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.03 + 1.75i)2-s + (−2.08 + 2.15i)3-s + (4.14 + 7.18i)4-s + (1.07 + 0.620i)5-s + (−10.1 + 2.88i)6-s + 15.0i·8-s + (−0.288 − 8.99i)9-s + (2.17 + 3.77i)10-s + (−6.07 + 3.50i)11-s + (−24.1 − 6.05i)12-s + 11.6·13-s + (−3.58 + 1.02i)15-s + (−9.79 + 16.9i)16-s + (−3.92 + 2.26i)17-s + (14.8 − 27.8i)18-s + (8.11 − 14.0i)19-s + ⋯
L(s)  = 1  + (1.51 + 0.876i)2-s + (−0.695 + 0.718i)3-s + (1.03 + 1.79i)4-s + (0.215 + 0.124i)5-s + (−1.68 + 0.480i)6-s + 1.88i·8-s + (−0.0321 − 0.999i)9-s + (0.217 + 0.377i)10-s + (−0.552 + 0.318i)11-s + (−2.01 − 0.504i)12-s + 0.895·13-s + (−0.238 + 0.0681i)15-s + (−0.611 + 1.05i)16-s + (−0.230 + 0.133i)17-s + (0.827 − 1.54i)18-s + (0.427 − 0.739i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.506 - 0.862i$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ -0.506 - 0.862i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.31089 + 2.29100i\)
\(L(\frac12)\) \(\approx\) \(1.31089 + 2.29100i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.08 - 2.15i)T \)
7 \( 1 \)
good2 \( 1 + (-3.03 - 1.75i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-1.07 - 0.620i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.07 - 3.50i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 11.6T + 169T^{2} \)
17 \( 1 + (3.92 - 2.26i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.11 + 14.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-22.1 - 12.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 9.49iT - 841T^{2} \)
31 \( 1 + (-14.3 - 24.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-16.5 + 28.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 67.1iT - 1.68e3T^{2} \)
43 \( 1 + 24.1T + 1.84e3T^{2} \)
47 \( 1 + (28.5 + 16.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (13.1 - 7.59i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (80.0 - 46.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (28.7 - 49.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (7.58 + 13.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 70.5iT - 5.04e3T^{2} \)
73 \( 1 + (38.3 + 66.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (63.6 - 110. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 74.2iT - 6.88e3T^{2} \)
89 \( 1 + (110. + 63.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 23.1T + 9.40e3T^{2} \)
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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

−13.31140797879442944450669967063, −12.37492507683446612090103211793, −11.40077217088802793475602898015, −10.40817296202777460369934520149, −8.916166459297586460759328185913, −7.32746893288923831286322879316, −6.30389364572874445887269950409, −5.38766649185740878175391243249, −4.45669302154587709099981823763, −3.20383740772059923987150952504, 1.42785495447295896291819307126, 3.00071819322850523332713097629, 4.63013303590405538663120351199, 5.67885533282129771377593293269, 6.47260797166334934402393542993, 8.059908761908241907836873970107, 9.951958014615563312240012138600, 11.11293393362739783426104198403, 11.51687021097547567688231217204, 12.70114381951833716044382575792

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