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Properties

Label 2-147-147.20-c1-0-10
Degree $2$
Conductor $147$
Sign $0.257 + 0.966i$
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  + (0.888 − 1.84i)2-s + (0.0997 + 1.72i)3-s + (−1.36 − 1.71i)4-s + (−0.654 − 2.86i)5-s + (3.27 + 1.35i)6-s + (2.64 + 0.113i)7-s + (−0.383 + 0.0875i)8-s + (−2.98 + 0.345i)9-s + (−5.87 − 1.33i)10-s + (0.921 − 1.91i)11-s + (2.82 − 2.53i)12-s + (−2.38 + 4.95i)13-s + (2.55 − 4.77i)14-s + (4.89 − 1.41i)15-s + (0.796 − 3.48i)16-s + (−1.65 + 2.06i)17-s + ⋯
L(s)  = 1  + (0.628 − 1.30i)2-s + (0.0576 + 0.998i)3-s + (−0.683 − 0.856i)4-s + (−0.292 − 1.28i)5-s + (1.33 + 0.551i)6-s + (0.999 + 0.0428i)7-s + (−0.135 + 0.0309i)8-s + (−0.993 + 0.115i)9-s + (−1.85 − 0.423i)10-s + (0.277 − 0.576i)11-s + (0.816 − 0.731i)12-s + (−0.661 + 1.37i)13-s + (0.683 − 1.27i)14-s + (1.26 − 0.366i)15-s + (0.199 − 0.872i)16-s + (−0.400 + 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.257 + 0.966i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20769 - 0.928011i\)
\(L(\frac12)\) \(\approx\) \(1.20769 - 0.928011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0997 - 1.72i)T \)
7 \( 1 + (-2.64 - 0.113i)T \)
good2 \( 1 + (-0.888 + 1.84i)T + (-1.24 - 1.56i)T^{2} \)
5 \( 1 + (0.654 + 2.86i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (-0.921 + 1.91i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.38 - 4.95i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (1.65 - 2.06i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 - 4.54iT - 19T^{2} \)
23 \( 1 + (6.09 - 4.86i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (-1.89 - 1.51i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + 1.91iT - 31T^{2} \)
37 \( 1 + (-1.26 + 1.58i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (-0.544 - 2.38i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-1.85 + 8.12i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.00 - 1.44i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (1.68 - 1.34i)T + (11.7 - 51.6i)T^{2} \)
59 \( 1 + (-2.06 + 9.02i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (8.10 + 6.46i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (1.90 - 1.52i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.210 + 0.436i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + (13.8 - 6.67i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.296 - 0.142i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−12.47707235055182576252014861278, −11.77006442936225424163859602486, −11.16192915437185604742065341229, −9.961199935847241872779726658733, −9.014059086830214831064371908746, −8.015811899021832871605976664982, −5.53233039924709230521502835948, −4.48425151007086961695734499511, −3.90367962272595137154372058283, −1.84141121182409568551603486774, 2.61390676770509024582529582793, 4.61843386963552802990868796604, 5.92749413248838485720956703267, 7.00219589646371392196196487378, 7.52007552631452097815693229766, 8.410348669698034623154191318941, 10.42732648744839948533637113321, 11.43344322508564382435394385677, 12.53438922194473119469108843521, 13.64286042753668966682995453440

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