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Properties

Label 2-168-168.101-c1-0-17
Degree $2$
Conductor $168$
Sign $0.766 - 0.642i$
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  + (0.998 + 1.00i)2-s + (1.26 − 1.18i)3-s + (−0.00659 + 1.99i)4-s + (−1.54 + 0.894i)5-s + (2.44 + 0.0881i)6-s + (2.63 − 0.230i)7-s + (−2.00 + 1.99i)8-s + (0.206 − 2.99i)9-s + (−2.44 − 0.658i)10-s + (−0.501 + 0.868i)11-s + (2.35 + 2.53i)12-s − 2.47·13-s + (2.86 + 2.41i)14-s + (−0.904 + 2.96i)15-s + (−3.99 − 0.0263i)16-s + (3.32 − 5.76i)17-s + ⋯
L(s)  = 1  + (0.705 + 0.708i)2-s + (0.730 − 0.682i)3-s + (−0.00329 + 0.999i)4-s + (−0.692 + 0.399i)5-s + (0.999 + 0.0360i)6-s + (0.996 − 0.0869i)7-s + (−0.710 + 0.703i)8-s + (0.0686 − 0.997i)9-s + (−0.772 − 0.208i)10-s + (−0.151 + 0.261i)11-s + (0.679 + 0.733i)12-s − 0.685·13-s + (0.764 + 0.644i)14-s + (−0.233 + 0.765i)15-s + (−0.999 − 0.00659i)16-s + (0.807 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71724 + 0.625000i\)
\(L(\frac12)\) \(\approx\) \(1.71724 + 0.625000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.998 - 1.00i)T \)
3 \( 1 + (-1.26 + 1.18i)T \)
7 \( 1 + (-2.63 + 0.230i)T \)
good5 \( 1 + (1.54 - 0.894i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.501 - 0.868i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
17 \( 1 + (-3.32 + 5.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.85 + 3.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.85 - 3.95i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.748T + 29T^{2} \)
31 \( 1 + (-2.87 - 1.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.22 + 1.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.01T + 41T^{2} \)
43 \( 1 - 9.19iT - 43T^{2} \)
47 \( 1 + (1.19 + 2.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.33 + 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.34 - 4.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.02 - 3.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.89 - 3.98i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.46iT - 71T^{2} \)
73 \( 1 + (5.68 + 3.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.53 - 4.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.65iT - 83T^{2} \)
89 \( 1 + (-7.39 - 12.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.75iT - 97T^{2} \)
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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

−13.14824577658165584647051053970, −11.91996968120601614772519181713, −11.54967471479600175075779006718, −9.628047435591488068970728559903, −8.252160798771402035828106220632, −7.60638251679159714299755953766, −6.91907593537799546043447471728, −5.28092163425708551475514547492, −3.96250409649817187281096956184, −2.56451970836623776720488112930, 2.11188037131759971268282908547, 3.80947849295740081158958077563, 4.53919762110128336297863305947, 5.78545036172197569130438493891, 7.893489665413150228044260716058, 8.539603392578470780454986392744, 10.01477565215201327251650852177, 10.63172094506065280543742059472, 11.85208933131357686534552566192, 12.51007475368583402931652105668

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