Properties

Degree $2$
Conductor $147$
Sign $0.0725 - 0.997i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 + 0.358i)2-s + (−0.5 − 0.866i)3-s + (0.914 + 1.58i)4-s + (−1.70 + 2.95i)5-s + 0.414·6-s − 1.58·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (1 + 1.73i)11-s + (0.914 − 1.58i)12-s + 2.58·13-s + 3.41·15-s + (−1.49 + 2.59i)16-s + (1.12 + 1.94i)17-s + (−0.207 − 0.358i)18-s + (1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (−0.146 + 0.253i)2-s + (−0.288 − 0.499i)3-s + (0.457 + 0.791i)4-s + (−0.763 + 1.32i)5-s + 0.169·6-s − 0.560·8-s + (−0.166 + 0.288i)9-s + (−0.223 − 0.387i)10-s + (0.301 + 0.522i)11-s + (0.263 − 0.457i)12-s + 0.717·13-s + 0.881·15-s + (−0.374 + 0.649i)16-s + (0.271 + 0.471i)17-s + (−0.0488 − 0.0845i)18-s + (0.324 − 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.0725 - 0.997i$
Motivic weight: \(1\)
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661837 + 0.615435i\)
\(L(\frac12)\) \(\approx\) \(0.661837 + 0.615435i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (0.207 - 0.358i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 + (-1.12 - 1.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.82 + 6.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + (-0.585 - 1.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.585 - 1.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.12 - 10.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.82 - 4.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 + (6.94 + 12.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.82 - 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 + (-7.12 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.58T + 97T^{2} \)
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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

−13.08894874646774176484905142560, −12.20280648265466440454787800514, −11.26939569636772156558244310609, −10.68946523169918932839344130740, −8.897148800663628897534960467783, −7.66455013513348093780612950244, −7.06756730521689905533640818552, −6.15976444849922651813775743876, −3.95109035381896103845163269446, −2.67266815677563568556195034027, 1.07459413278504544289171169784, 3.63845703396218886885321011549, 5.07039082709885831225379431904, 5.97371222579920045515647232408, 7.65256890414441409205250270468, 8.974662692645919751940224408015, 9.623971350396168809469681098609, 11.07745381072034168689527576288, 11.53942670230990264215599362307, 12.57558587618559825438131320110

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