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Properties

Label 2-168-56.3-c1-0-9
Degree $2$
Conductor $168$
Sign $0.835 + 0.549i$
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  + (−0.297 + 1.38i)2-s + (−0.866 − 0.5i)3-s + (−1.82 − 0.821i)4-s + (−1.44 − 2.49i)5-s + (0.948 − 1.04i)6-s + (2.63 + 0.194i)7-s + (1.67 − 2.27i)8-s + (0.499 + 0.866i)9-s + (3.88 − 1.25i)10-s + (2.91 − 5.04i)11-s + (1.16 + 1.62i)12-s − 1.04·13-s + (−1.05 + 3.59i)14-s + 2.88i·15-s + (2.64 + 2.99i)16-s + (−5.91 − 3.41i)17-s + ⋯
L(s)  = 1  + (−0.210 + 0.977i)2-s + (−0.499 − 0.288i)3-s + (−0.911 − 0.410i)4-s + (−0.644 − 1.11i)5-s + (0.387 − 0.428i)6-s + (0.997 + 0.0733i)7-s + (0.593 − 0.805i)8-s + (0.166 + 0.288i)9-s + (1.22 − 0.395i)10-s + (0.878 − 1.52i)11-s + (0.337 + 0.468i)12-s − 0.290·13-s + (−0.281 + 0.959i)14-s + 0.744i·15-s + (0.662 + 0.749i)16-s + (−1.43 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718742 - 0.215148i\)
\(L(\frac12)\) \(\approx\) \(0.718742 - 0.215148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.297 - 1.38i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-2.63 - 0.194i)T \)
good5 \( 1 + (1.44 + 2.49i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.91 + 5.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
17 \( 1 + (5.91 + 3.41i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.589 - 0.340i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.85 + 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.61iT - 29T^{2} \)
31 \( 1 + (-1.91 + 3.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.06 + 1.19i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.19iT - 41T^{2} \)
43 \( 1 - 1.34T + 43T^{2} \)
47 \( 1 + (-5.52 - 9.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.99 + 4.03i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.81 - 3.93i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.63 + 2.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.65 - 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.08iT - 71T^{2} \)
73 \( 1 + (-4.88 - 2.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.9 + 6.32i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.482iT - 83T^{2} \)
89 \( 1 + (-10.7 + 6.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.63iT - 97T^{2} \)
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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

−12.81310097309295223967360557228, −11.67243647452499682103712289500, −10.93178143206181267886539949256, −9.069418811783590152973947703454, −8.596229455069735174298640573691, −7.56653896904097926186311583356, −6.33128430032706556247289792031, −5.12142898265596983022459079354, −4.29941058162651676578244070232, −0.881807971014678891294162207777, 2.09556843334520905767543930009, 3.94723382632786933718750051933, 4.72573805715045976545324830991, 6.71748794144035248492032995610, 7.76127666646308628475730739173, 9.083262485005410685364419365370, 10.23860415986680213961407075345, 10.99126521882961340862699373974, 11.66151945527554853545357408006, 12.43566402486987933433139497688

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