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Properties

Label 2-147-7.4-c3-0-5
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  + (2.27 + 3.94i)2-s + (−1.5 + 2.59i)3-s + (−6.38 + 11.0i)4-s + (8.93 + 15.4i)5-s − 13.6·6-s − 21.6·8-s + (−4.5 − 7.79i)9-s + (−40.7 + 70.5i)10-s + (5.69 − 9.86i)11-s + (−19.1 − 33.1i)12-s + 13.0·13-s − 53.6·15-s + (1.62 + 2.81i)16-s + (26.6 − 46.1i)17-s + (20.5 − 35.5i)18-s + (−21.2 − 36.7i)19-s + ⋯
L(s)  = 1  + (0.805 + 1.39i)2-s + (−0.288 + 0.499i)3-s + (−0.797 + 1.38i)4-s + (0.799 + 1.38i)5-s − 0.930·6-s − 0.958·8-s + (−0.166 − 0.288i)9-s + (−1.28 + 2.23i)10-s + (0.156 − 0.270i)11-s + (−0.460 − 0.797i)12-s + 0.279·13-s − 0.922·15-s + (0.0254 + 0.0440i)16-s + (0.379 − 0.658i)17-s + (0.268 − 0.465i)18-s + (−0.256 − 0.443i)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

\(L(2)\) \(\approx\) \(0.157897 - 2.48813i\)
\(L(\frac12)\) \(\approx\) \(0.157897 - 2.48813i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.27 - 3.94i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-8.93 - 15.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-5.69 + 9.86i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 13.0T + 2.19e3T^{2} \)
17 \( 1 + (-26.6 + 46.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (21.2 + 36.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (76.0 + 131. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 186.T + 2.43e4T^{2} \)
31 \( 1 + (78.9 - 136. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (1.87 + 3.24i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 39.3T + 6.89e4T^{2} \)
43 \( 1 - 429.T + 7.95e4T^{2} \)
47 \( 1 + (-10.5 - 18.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (182. - 316. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (113. - 196. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-325. - 564. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (72.7 - 125. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 368.T + 3.57e5T^{2} \)
73 \( 1 + (-304. + 527. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (455. + 788. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 327.T + 5.71e5T^{2} \)
89 \( 1 + (18.8 + 32.5i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 722.T + 9.12e5T^{2} \)
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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

−13.71140763072207944568851996964, −12.36686976128796343924697200358, −10.94242109098095606710341393383, −10.16413447818936065364710759085, −8.770564804100162804694753639875, −7.32377530116819357787820910970, −6.43321156440088790268761067141, −5.75154108682978443626818290439, −4.41919504632536102489123976200, −2.94694757304272117330287773144, 1.08665066379249628164541900338, 2.05059246795715220688861912839, 3.95061587395213310660861486420, 5.15060713154179506295315021222, 6.02457644356199206971535000419, 8.015545446125984816370305027448, 9.345189397406917444085569703596, 10.18823811110510683603012033854, 11.38032193992950744432530759887, 12.35140627928324777582987127803

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