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Properties

Label 2-147-7.4-c5-0-26
Degree $2$
Conductor $147$
Sign $-0.968 - 0.250i$
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  + (−0.5 − 0.866i)2-s + (−4.5 + 7.79i)3-s + (15.5 − 26.8i)4-s + (−17 − 29.4i)5-s + 9·6-s − 63·8-s + (−40.5 − 70.1i)9-s + (−16.9 + 29.4i)10-s + (170 − 294. i)11-s + (139.5 + 241. i)12-s − 454·13-s + 306·15-s + (−464.5 − 804. i)16-s + (−399 + 691. i)17-s + (−40.5 + 70.1i)18-s + (446 + 772. i)19-s + ⋯
L(s)  = 1  + (−0.0883 − 0.153i)2-s + (−0.288 + 0.499i)3-s + (0.484 − 0.838i)4-s + (−0.304 − 0.526i)5-s + 0.102·6-s − 0.348·8-s + (−0.166 − 0.288i)9-s + (−0.0537 + 0.0931i)10-s + (0.423 − 0.733i)11-s + (0.279 + 0.484i)12-s − 0.745·13-s + 0.351·15-s + (−0.453 − 0.785i)16-s + (−0.334 + 0.579i)17-s + (−0.0294 + 0.0510i)18-s + (0.283 + 0.490i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.968 - 0.250i$
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: \((2,\ 147,\ (\ :5/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3548136278\)
\(L(\frac12)\) \(\approx\) \(0.3548136278\)
\(L(\frac{7}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (17 + 29.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-170 + 294. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 454T + 3.71e5T^{2} \)
17 \( 1 + (399 - 691. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-446 - 772. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.59e3 - 2.76e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 8.24e3T + 2.05e7T^{2} \)
31 \( 1 + (1.24e3 - 2.16e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (4.89e3 + 8.48e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.98e4T + 1.15e8T^{2} \)
43 \( 1 + 1.72e4T + 1.47e8T^{2} \)
47 \( 1 + (-4.46e3 - 7.73e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (75 - 129. i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.11e4 - 3.67e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-7.37e3 - 1.27e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-838 + 1.45e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.45e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.91e4 + 6.78e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.13e3 - 1.96e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 3.77e4T + 3.93e9T^{2} \)
89 \( 1 + (5.86e4 + 1.01e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.00e4T + 8.58e9T^{2} \)
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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

−11.47433714165559871693328137385, −10.64530021488145323892153373047, −9.626691942940905043951850174883, −8.722738312335438389974241508886, −7.19629727661128669694281016612, −5.90440081564089558128575018383, −5.00337794278833868111779082772, −3.49124243158454700518046662289, −1.61001454607709821291771954059, −0.12097618969151693032224129828, 2.07938356322734420877207094939, 3.36170736378964369911049356106, 4.97580495163268527437942507653, 6.86287296352405389711436577892, 7.03429203284902412971283531114, 8.275079812084057674235200964286, 9.548853008191748536800693788864, 11.00168931478824423519784279495, 11.71426069185043276902069386303, 12.54321727882313697855467309440

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