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Properties

Label 2-147-147.104-c1-0-14
Degree $2$
Conductor $147$
Sign $0.523 + 0.851i$
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  + (0.708 + 0.565i)2-s + (0.562 − 1.63i)3-s + (−0.262 − 1.14i)4-s + (−3.11 − 1.50i)5-s + (1.32 − 0.842i)6-s + (1.97 + 1.76i)7-s + (1.25 − 2.59i)8-s + (−2.36 − 1.84i)9-s + (−1.36 − 2.82i)10-s + (4.39 + 3.50i)11-s + (−2.02 − 0.217i)12-s + (0.177 + 0.141i)13-s + (0.402 + 2.36i)14-s + (−4.21 + 4.26i)15-s + (0.230 − 0.111i)16-s + (0.323 − 1.41i)17-s + ⋯
L(s)  = 1  + (0.501 + 0.399i)2-s + (0.325 − 0.945i)3-s + (−0.131 − 0.574i)4-s + (−1.39 − 0.671i)5-s + (0.540 − 0.344i)6-s + (0.745 + 0.666i)7-s + (0.441 − 0.917i)8-s + (−0.788 − 0.614i)9-s + (−0.430 − 0.894i)10-s + (1.32 + 1.05i)11-s + (−0.585 − 0.0626i)12-s + (0.0492 + 0.0392i)13-s + (0.107 + 0.631i)14-s + (−1.08 + 1.10i)15-s + (0.0577 − 0.0277i)16-s + (0.0784 − 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.523 + 0.851i$
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: \((2,\ 147,\ (\ :1/2),\ 0.523 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17104 - 0.654725i\)
\(L(\frac12)\) \(\approx\) \(1.17104 - 0.654725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.562 + 1.63i)T \)
7 \( 1 + (-1.97 - 1.76i)T \)
good2 \( 1 + (-0.708 - 0.565i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (3.11 + 1.50i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-4.39 - 3.50i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.177 - 0.141i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.323 + 1.41i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 5.02iT - 19T^{2} \)
23 \( 1 + (-3.57 + 0.815i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.768 - 0.175i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 4.63iT - 31T^{2} \)
37 \( 1 + (0.757 - 3.31i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (5.72 + 2.75i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (1.08 - 0.521i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (3.10 - 3.89i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (7.48 - 1.70i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-9.29 + 4.47i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.684 + 0.156i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 4.45T + 67T^{2} \)
71 \( 1 + (4.05 - 0.925i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.09 + 1.66i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 8.36T + 79T^{2} \)
83 \( 1 + (-7.02 - 8.80i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-8.02 - 10.0i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−12.75846124939596764689973377145, −12.10130324239492793251158454253, −11.39574663960666181114557980577, −9.513563674553827276252500236445, −8.494346903280204564257270772588, −7.53276359152396056594789194674, −6.49097055566059714721335947248, −5.05922437525430306332587345018, −3.94883946369733684727335936566, −1.44337991929851478351338986743, 3.22944302818210124701877602328, 3.86564805849188296128506790385, 4.84200615743306971670521527692, 7.01517287558906011586756180293, 8.164360781683794532503928310692, 8.856961491011595238043908929504, 10.67395603884567008160956970601, 11.34064444606006212798268925958, 11.76449004604147979987366473133, 13.39164374418831832215217583643

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