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Properties

Label 2-147-147.62-c1-0-15
Degree $2$
Conductor $147$
Sign $-0.779 + 0.626i$
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  + (0.399 − 0.0910i)2-s + (−0.596 − 1.62i)3-s + (−1.65 + 0.795i)4-s + (−1.77 − 2.22i)5-s + (−0.386 − 0.594i)6-s + (−2.45 − 0.986i)7-s + (−1.22 + 0.977i)8-s + (−2.28 + 1.93i)9-s + (−0.909 − 0.725i)10-s + (3.98 − 0.909i)11-s + (2.27 + 2.21i)12-s + (4.69 − 1.07i)13-s + (−1.06 − 0.170i)14-s + (−2.55 + 4.20i)15-s + (1.88 − 2.36i)16-s + (−4.81 − 2.31i)17-s + ⋯
L(s)  = 1  + (0.282 − 0.0643i)2-s + (−0.344 − 0.938i)3-s + (−0.825 + 0.397i)4-s + (−0.792 − 0.993i)5-s + (−0.157 − 0.242i)6-s + (−0.927 − 0.372i)7-s + (−0.433 + 0.345i)8-s + (−0.763 + 0.646i)9-s + (−0.287 − 0.229i)10-s + (1.20 − 0.274i)11-s + (0.657 + 0.638i)12-s + (1.30 − 0.297i)13-s + (−0.285 − 0.0454i)14-s + (−0.660 + 1.08i)15-s + (0.471 − 0.590i)16-s + (−1.16 − 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.779 + 0.626i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.206895 - 0.587583i\)
\(L(\frac12)\) \(\approx\) \(0.206895 - 0.587583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.596 + 1.62i)T \)
7 \( 1 + (2.45 + 0.986i)T \)
good2 \( 1 + (-0.399 + 0.0910i)T + (1.80 - 0.867i)T^{2} \)
5 \( 1 + (1.77 + 2.22i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (-3.98 + 0.909i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-4.69 + 1.07i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (4.81 + 2.31i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + 4.40iT - 19T^{2} \)
23 \( 1 + (0.270 + 0.561i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (2.44 - 5.08i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + 0.670iT - 31T^{2} \)
37 \( 1 + (6.37 + 3.06i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-4.48 - 5.62i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-3.55 + 4.46i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.622 - 2.72i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (1.18 + 2.45i)T + (-33.0 + 41.4i)T^{2} \)
59 \( 1 + (-3.41 + 4.28i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (1.93 - 4.01i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 - 3.62T + 67T^{2} \)
71 \( 1 + (2.32 + 4.82i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (10.7 + 2.46i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + (-3.03 + 13.3i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-1.11 + 4.87i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + 12.2iT - 97T^{2} \)
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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

−12.85344097650252068837450472152, −11.93144256007431326667523762569, −11.06526214691857710709521342016, −9.008986074293040959625003418953, −8.713408998441966040018066701062, −7.30098962989104800821881847324, −6.14212539317655550686275302219, −4.65414938297832591123233663693, −3.49476735215379814442355400705, −0.62243394954893945587830695121, 3.63891662636888340800855946365, 4.06769978015225713134900698117, 5.92222376827502636599791589933, 6.59438883555615127129141364949, 8.614747626763533383349692126199, 9.404008577422356880367284874339, 10.43111435400049883734558744768, 11.33841920206786457624881114487, 12.33487795675338976089576809333, 13.64667184414640946400199853054

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