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Properties

Label 2-147-21.5-c3-0-23
Degree $2$
Conductor $147$
Sign $0.896 + 0.443i$
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  + (2.24 + 1.29i)2-s + (−2.59 + 4.50i)3-s + (−0.652 − 1.13i)4-s + (8.05 − 13.9i)5-s + (−11.6 + 6.73i)6-s − 24.0i·8-s + (−13.5 − 23.3i)9-s + (36.1 − 20.8i)10-s + (30.8 − 17.7i)11-s + (6.78 − 0.00678i)12-s − 7.40i·13-s + (41.9 + 72.4i)15-s + (25.9 − 44.9i)16-s + (14.4 + 25.0i)17-s + (−0.139 − 69.8i)18-s + (−30.4 − 17.5i)19-s + ⋯
L(s)  = 1  + (0.792 + 0.457i)2-s + (−0.499 + 0.866i)3-s + (−0.0815 − 0.141i)4-s + (0.720 − 1.24i)5-s + (−0.791 + 0.458i)6-s − 1.06i·8-s + (−0.501 − 0.865i)9-s + (1.14 − 0.659i)10-s + (0.845 − 0.487i)11-s + (0.163 − 0.000163i)12-s − 0.158i·13-s + (0.722 + 1.24i)15-s + (0.405 − 0.701i)16-s + (0.206 + 0.357i)17-s + (−0.00182 − 0.914i)18-s + (−0.367 − 0.212i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.896 + 0.443i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.09539 - 0.489765i\)
\(L(\frac12)\) \(\approx\) \(2.09539 - 0.489765i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.59 - 4.50i)T \)
7 \( 1 \)
good2 \( 1 + (-2.24 - 1.29i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-8.05 + 13.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.8 + 17.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 7.40iT - 2.19e3T^{2} \)
17 \( 1 + (-14.4 - 25.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (30.4 + 17.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-48.0 - 27.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 68.1iT - 2.43e4T^{2} \)
31 \( 1 + (-154. + 89.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-116. + 202. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 370.T + 6.89e4T^{2} \)
43 \( 1 + 187.T + 7.95e4T^{2} \)
47 \( 1 + (-87.3 + 151. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (235. - 136. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-48.4 - 83.8i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-333. - 192. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-509. - 881. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 125. iT - 3.57e5T^{2} \)
73 \( 1 + (195. - 112. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-532. + 921. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 601.T + 5.71e5T^{2} \)
89 \( 1 + (752. - 1.30e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 327. iT - 9.12e5T^{2} \)
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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

−12.72441393194214005708242811103, −11.68984090188084014654347686871, −10.32144657487452734346618986283, −9.434377380774029148826219389943, −8.666529051629385367925221141232, −6.52821240802876434170075800466, −5.63397659445375165247554496612, −4.86461294355747098445027483595, −3.80389210546529201451989698799, −0.942353557475698012540517670709, 1.94210943118042290916068915356, 3.12663740848558179236137857453, 4.82991481555179421271930481502, 6.20651655409274247134400952024, 6.95650308714111600698455128459, 8.297315336375531635543219542646, 9.895656955020548322480027161397, 11.03638053664503964920134297702, 11.78676434875255605775716656465, 12.63962224751236959914900012355

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