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Properties

Label 2-147-7.4-c5-0-25
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  + (−1.37 − 2.37i)2-s + (4.5 − 7.79i)3-s + (12.2 − 21.1i)4-s + (29.1 + 50.5i)5-s − 24.7·6-s − 155.·8-s + (−40.5 − 70.1i)9-s + (80.2 − 138. i)10-s + (−8.71 + 15.0i)11-s + (−110. − 190. i)12-s − 889.·13-s + 525.·15-s + (−178. − 308. i)16-s + (513. − 889. i)17-s + (−111. + 192. i)18-s + (−869. − 1.50e3i)19-s + ⋯
L(s)  = 1  + (−0.242 − 0.420i)2-s + (0.288 − 0.499i)3-s + (0.381 − 0.661i)4-s + (0.522 + 0.904i)5-s − 0.280·6-s − 0.856·8-s + (−0.166 − 0.288i)9-s + (0.253 − 0.439i)10-s + (−0.0217 + 0.0376i)11-s + (−0.220 − 0.381i)12-s − 1.46·13-s + 0.602·15-s + (−0.173 − 0.301i)16-s + (0.430 − 0.746i)17-s + (−0.0809 + 0.140i)18-s + (−0.552 − 0.957i)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

\(L(3)\) \(\approx\) \(1.226351768\)
\(L(\frac12)\) \(\approx\) \(1.226351768\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.37 + 2.37i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-29.1 - 50.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (8.71 - 15.0i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 889.T + 3.71e5T^{2} \)
17 \( 1 + (-513. + 889. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (869. + 1.50e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.96e3 + 3.40e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 5.63e3T + 2.05e7T^{2} \)
31 \( 1 + (1.54e3 - 2.68e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.51e3 + 4.35e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.83e4T + 1.15e8T^{2} \)
43 \( 1 + 1.63e3T + 1.47e8T^{2} \)
47 \( 1 + (4.80e3 + 8.31e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.16e4 + 2.01e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.80e3 - 3.12e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.14e4 - 1.98e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.35e4 - 4.07e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.59e3T + 1.80e9T^{2} \)
73 \( 1 + (-2.96e3 + 5.13e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.42e4 - 7.66e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 9.58e4T + 3.93e9T^{2} \)
89 \( 1 + (2.32e4 + 4.02e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 7.59e4T + 8.58e9T^{2} \)
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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

−11.65795208804143195692399580628, −10.39754567553760308304574113805, −9.948875825113031802585893931943, −8.669227509482487410975891443721, −7.07690440206091156886222192034, −6.51151774398990261191186259113, −5.06574526085432483642138370971, −2.81465718732409702506243497772, −2.15694558860676772556769789913, −0.38892228681373068113390755483, 1.93027652560405659351694265794, 3.48683772359282345589540075745, 4.92712954383861458374969040833, 6.13261458695421887491243551211, 7.61166578529036843484819667717, 8.403104705221070633884655646624, 9.439625051052719463334890828476, 10.28450796188846584782869603393, 11.89523144101434572779430449204, 12.49202621023790703216079888887

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