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Properties

Label 2-147-21.20-c5-0-54
Degree $2$
Conductor $147$
Sign $-0.292 - 0.956i$
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  − 8.31i·2-s + (−14.0 + 6.83i)3-s − 37.1·4-s + 28.6·5-s + (56.8 + 116. i)6-s + 42.3i·8-s + (149. − 191. i)9-s − 238. i·10-s − 596. i·11-s + (519. − 253. i)12-s + 602. i·13-s + (−401. + 196. i)15-s − 834.·16-s − 154.·17-s + (−1.59e3 − 1.24e3i)18-s − 2.86e3i·19-s + ⋯
L(s)  = 1  − 1.46i·2-s + (−0.898 + 0.438i)3-s − 1.15·4-s + 0.513·5-s + (0.644 + 1.32i)6-s + 0.234i·8-s + (0.615 − 0.788i)9-s − 0.754i·10-s − 1.48i·11-s + (1.04 − 0.508i)12-s + 0.988i·13-s + (−0.461 + 0.225i)15-s − 0.815·16-s − 0.129·17-s + (−1.15 − 0.904i)18-s − 1.82i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.292 - 0.956i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.292 - 0.956i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4448976381\)
\(L(\frac12)\) \(\approx\) \(0.4448976381\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (14.0 - 6.83i)T \)
7 \( 1 \)
good2 \( 1 + 8.31iT - 32T^{2} \)
5 \( 1 - 28.6T + 3.12e3T^{2} \)
11 \( 1 + 596. iT - 1.61e5T^{2} \)
13 \( 1 - 602. iT - 3.71e5T^{2} \)
17 \( 1 + 154.T + 1.41e6T^{2} \)
19 \( 1 + 2.86e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.78e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.85e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.22e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.04e4T + 6.93e7T^{2} \)
41 \( 1 - 6.92e3T + 1.15e8T^{2} \)
43 \( 1 - 3.80e3T + 1.47e8T^{2} \)
47 \( 1 + 1.57e4T + 2.29e8T^{2} \)
53 \( 1 + 4.78e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.34e4T + 7.14e8T^{2} \)
61 \( 1 - 4.51e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.95e4T + 1.35e9T^{2} \)
71 \( 1 + 3.59e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.67e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.05e3T + 3.07e9T^{2} \)
83 \( 1 + 3.62e4T + 3.93e9T^{2} \)
89 \( 1 + 7.63e4T + 5.58e9T^{2} \)
97 \( 1 - 5.19e4iT - 8.58e9T^{2} \)
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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

−11.35035020702420636892183506033, −10.76608696163614982759778747682, −9.610647493784718463015099424839, −8.984577748924264781114121989654, −6.84834199674347340307451786936, −5.64659973546707648670307723721, −4.34376699002012518907804570604, −3.11784866279452666634309407268, −1.49121952732824441253704157833, −0.16910399054725377919946279424, 1.88395503337498834937093673682, 4.55459367664281529981405112023, 5.58766312545660923366958581350, 6.36342127327618046486641619448, 7.37817526650885764662522995327, 8.177813791874548552481882897774, 9.770667765900037359218663946792, 10.64566927199361619040653815638, 12.20239821447018345977687478740, 12.84085038867222105322323464136

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