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Properties

Label 2-147-147.104-c1-0-3
Degree $2$
Conductor $147$
Sign $0.433 + 0.901i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.95 − 1.56i)2-s + (1.73 − 0.0781i)3-s + (0.951 + 4.16i)4-s + (−0.242 − 0.116i)5-s + (−3.51 − 2.54i)6-s + (2.60 + 0.481i)7-s + (2.47 − 5.13i)8-s + (2.98 − 0.270i)9-s + (0.292 + 0.607i)10-s + (2.92 + 2.33i)11-s + (1.97 + 7.13i)12-s + (−4.15 − 3.31i)13-s + (−4.34 − 5.00i)14-s + (−0.428 − 0.183i)15-s + (−5.16 + 2.48i)16-s + (0.256 − 1.12i)17-s + ⋯
L(s)  = 1  + (−1.38 − 1.10i)2-s + (0.998 − 0.0451i)3-s + (0.475 + 2.08i)4-s + (−0.108 − 0.0522i)5-s + (−1.43 − 1.04i)6-s + (0.983 + 0.181i)7-s + (0.874 − 1.81i)8-s + (0.995 − 0.0901i)9-s + (0.0925 + 0.192i)10-s + (0.881 + 0.703i)11-s + (0.569 + 2.06i)12-s + (−1.15 − 0.919i)13-s + (−1.16 − 1.33i)14-s + (−0.110 − 0.0472i)15-s + (−1.29 + 0.621i)16-s + (0.0621 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.433 + 0.901i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.433 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718747 - 0.451743i\)
\(L(\frac12)\) \(\approx\) \(0.718747 - 0.451743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 + 0.0781i)T \)
7 \( 1 + (-2.60 - 0.481i)T \)
good2 \( 1 + (1.95 + 1.56i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (0.242 + 0.116i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.92 - 2.33i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (4.15 + 3.31i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.256 + 1.12i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 2.04iT - 19T^{2} \)
23 \( 1 + (5.12 - 1.16i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-4.08 - 0.932i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 6.91iT - 31T^{2} \)
37 \( 1 + (1.90 - 8.32i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-0.0133 - 0.00643i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.419 + 0.202i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-6.73 + 8.44i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (6.59 - 1.50i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (9.03 - 4.35i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (9.35 + 2.13i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 1.25T + 67T^{2} \)
71 \( 1 + (13.9 - 3.18i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (3.13 - 2.49i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 2.82T + 79T^{2} \)
83 \( 1 + (2.80 + 3.51i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.44 - 1.80i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 2.81iT - 97T^{2} \)
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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

−12.21638007456718437925595605872, −12.03244091956498871376672478135, −10.48468932135696850146315014485, −9.817917899151271075409819640672, −8.828036749720095662883715691788, −8.025579996379112688272862090680, −7.21436638459130427509131223161, −4.50760352388890182920776962522, −2.88802731217797671317387012021, −1.66060655812278351465315979367, 1.77264683538807771712933060735, 4.28084598329985614258793978224, 6.07932932301345865215287893608, 7.39480444817131043057556824174, 7.945479207033334408304553725337, 8.980988457127131446529045697641, 9.639262954668740106379529385521, 10.78064240468147479329387255713, 12.04698385237093749246608557574, 13.98270889120030855595442871838

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