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Properties

Label 2-147-7.4-c5-0-15
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  + (−5.11 − 8.85i)2-s + (4.5 − 7.79i)3-s + (−36.2 + 62.8i)4-s + (11.8 + 20.5i)5-s − 92.0·6-s + 414.·8-s + (−40.5 − 70.1i)9-s + (121. − 210. i)10-s + (−232. + 403. i)11-s + (326. + 565. i)12-s + 1.01e3·13-s + 213.·15-s + (−957. − 1.65e3i)16-s + (280. − 486. i)17-s + (−414. + 717. i)18-s + (−693. − 1.20e3i)19-s + ⋯
L(s)  = 1  + (−0.903 − 1.56i)2-s + (0.288 − 0.499i)3-s + (−1.13 + 1.96i)4-s + (0.212 + 0.367i)5-s − 1.04·6-s + 2.28·8-s + (−0.166 − 0.288i)9-s + (0.383 − 0.665i)10-s + (−0.580 + 1.00i)11-s + (0.654 + 1.13i)12-s + 1.67·13-s + 0.245·15-s + (−0.935 − 1.61i)16-s + (0.235 − 0.408i)17-s + (−0.301 + 0.521i)18-s + (−0.440 − 0.763i)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: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.052412797\)
\(L(\frac12)\) \(\approx\) \(1.052412797\)
\(L(\frac{7}{2})\) 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 \( 1 + (5.11 + 8.85i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-11.8 - 20.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (232. - 403. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 1.01e3T + 3.71e5T^{2} \)
17 \( 1 + (-280. + 486. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (693. + 1.20e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (2.05e3 + 3.56e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 2.38e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.47e3 + 2.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.95e3 - 8.58e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 4.47e3T + 1.15e8T^{2} \)
43 \( 1 - 5.18e3T + 1.47e8T^{2} \)
47 \( 1 + (1.56e3 + 2.70e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (570. - 988. i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.37e4 + 2.38e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.05e4 + 1.82e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.77e4 + 4.81e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.07e3T + 1.80e9T^{2} \)
73 \( 1 + (8.38e3 - 1.45e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.42e3 - 4.19e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.01e4T + 3.93e9T^{2} \)
89 \( 1 + (3.12e4 + 5.41e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 6.36e4T + 8.58e9T^{2} \)
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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

−11.46447019241801204306821773389, −10.64151187857672344600850050671, −9.793758036504880033244410474658, −8.700150711315663389742166283506, −7.905906258094655659614503198236, −6.47897762938852464969642430730, −4.30261160042549693159129301285, −2.88507736115406102639415293585, −1.94483381893942027133782249754, −0.54729771755197061543776263797, 1.18639935162307989511689772319, 3.79577306955930683834142523529, 5.54555564741541791102241724933, 6.02149145143438953526290768287, 7.63276852745452037266611405024, 8.461969202012365247862509669241, 9.103192578622394364126833188663, 10.21622869522160308229197000928, 11.13709753126570858894956685514, 13.14635304052177113524246113008

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