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Properties

Label 2-168-56.19-c1-0-9
Degree $2$
Conductor $168$
Sign $0.977 - 0.212i$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
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  + (−0.321 + 1.37i)2-s + (0.866 − 0.5i)3-s + (−1.79 − 0.886i)4-s + (1.25 − 2.16i)5-s + (0.409 + 1.35i)6-s + (1.36 − 2.26i)7-s + (1.79 − 2.18i)8-s + (0.499 − 0.866i)9-s + (2.58 + 2.42i)10-s + (2.83 + 4.91i)11-s + (−1.99 + 0.128i)12-s − 5.31·13-s + (2.68 + 2.60i)14-s − 2.50i·15-s + (2.42 + 3.17i)16-s + (−0.393 + 0.227i)17-s + ⋯
L(s)  = 1  + (−0.227 + 0.973i)2-s + (0.499 − 0.288i)3-s + (−0.896 − 0.443i)4-s + (0.559 − 0.969i)5-s + (0.167 + 0.552i)6-s + (0.515 − 0.857i)7-s + (0.635 − 0.771i)8-s + (0.166 − 0.288i)9-s + (0.816 + 0.765i)10-s + (0.855 + 1.48i)11-s + (−0.576 + 0.0370i)12-s − 1.47·13-s + (0.717 + 0.696i)14-s − 0.646i·15-s + (0.606 + 0.794i)16-s + (−0.0955 + 0.0551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.977 - 0.212i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.977 - 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21404 + 0.130268i\)
\(L(\frac12)\) \(\approx\) \(1.21404 + 0.130268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.321 - 1.37i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-1.36 + 2.26i)T \)
good5 \( 1 + (-1.25 + 2.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.83 - 4.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.31T + 13T^{2} \)
17 \( 1 + (0.393 - 0.227i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.19 + 1.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.43 - 2.56i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.57iT - 29T^{2} \)
31 \( 1 + (-3.00 - 5.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.80 + 4.50i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.65iT - 41T^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 + (0.478 - 0.829i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.41 - 3.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.76 - 5.06i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.50 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.65 - 8.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.35iT - 71T^{2} \)
73 \( 1 + (-5.93 + 3.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.71 + 4.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.96iT - 83T^{2} \)
89 \( 1 + (-5.91 - 3.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.71iT - 97T^{2} \)
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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

−12.95311571782810244955272862484, −12.30710142834189814422099189637, −10.40574543548432534870910733206, −9.431764386343007773754796865557, −8.796776526945597875405425801873, −7.41387195559321340748863946379, −6.91675034308258275801277039240, −5.09870952118493913560410267314, −4.37706881657785205073447329229, −1.55779170921265360376648425862, 2.23839602004584759180305868252, 3.20840213741879733696587244230, 4.82264402401981571834606172697, 6.32130755081903175766185151460, 7.998847893492795667748125128753, 8.929012982301880863357105540723, 9.793944497014458397339577503631, 10.79174918927856175330181987389, 11.57494354526810911436523824432, 12.59659785802688249208378666677

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