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Properties

Label 2-147-7.2-c5-0-22
Degree $2$
Conductor $147$
Sign $0.386 + 0.922i$
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  + (3 − 5.19i)2-s + (−4.5 − 7.79i)3-s + (−2 − 3.46i)4-s + (−3 + 5.19i)5-s − 54·6-s + 168·8-s + (−40.5 + 70.1i)9-s + (18 + 31.1i)10-s + (282 + 488. i)11-s + (−18.0 + 31.1i)12-s + 638·13-s + 54·15-s + (568 − 983. i)16-s + (−441 − 763. i)17-s + (243 + 420. i)18-s + (278 − 481. i)19-s + ⋯
L(s)  = 1  + (0.530 − 0.918i)2-s + (−0.288 − 0.499i)3-s + (−0.0625 − 0.108i)4-s + (−0.0536 + 0.0929i)5-s − 0.612·6-s + 0.928·8-s + (−0.166 + 0.288i)9-s + (0.0569 + 0.0985i)10-s + (0.702 + 1.21i)11-s + (−0.0360 + 0.0625i)12-s + 1.04·13-s + 0.0619·15-s + (0.554 − 0.960i)16-s + (−0.370 − 0.641i)17-s + (0.176 + 0.306i)18-s + (0.176 − 0.305i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.799290290\)
\(L(\frac12)\) \(\approx\) \(2.799290290\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3 + 5.19i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (3 - 5.19i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-282 - 488. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 638T + 3.71e5T^{2} \)
17 \( 1 + (441 + 763. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-278 + 481. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-420 + 727. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 4.63e3T + 2.05e7T^{2} \)
31 \( 1 + (2.20e3 + 3.81e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.20e3 + 2.08e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 6.87e3T + 1.15e8T^{2} \)
43 \( 1 - 9.64e3T + 1.47e8T^{2} \)
47 \( 1 + (-9.33e3 + 1.61e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.68e4 + 2.92e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-9.04e3 - 1.56e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.98e4 - 3.44e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.15e4 - 1.99e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 4.24e3T + 1.80e9T^{2} \)
73 \( 1 + (-2.05e4 - 3.56e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.09e4 - 1.89e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 8.24e4T + 3.93e9T^{2} \)
89 \( 1 + (-4.70e4 + 8.14e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 4.94e4T + 8.58e9T^{2} \)
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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

−11.90778884714561719284705689024, −11.29464208594509385140633168421, −10.27498540426477541396479886761, −8.960664635691604812565763139021, −7.49777515162677680518887614385, −6.63027983461981637133109721718, −4.98185980028052284672400834756, −3.81682250413544175813984633584, −2.38050390290165697923108390506, −1.14587030938156797064657435408, 1.13870219126878295669603293487, 3.53957087168519591641235408804, 4.68743937478067300448234885841, 5.97821623296091101697321192634, 6.48155236667716137173350538658, 8.071052544266762428696511892278, 9.022008078519414652505211453499, 10.54782916278907889013904056008, 11.12968398966023896321589912350, 12.45372071946020634920031330111

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