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Properties

Label 2-168-168.101-c1-0-7
Degree $2$
Conductor $168$
Sign $0.849 - 0.528i$
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  + (−1.26 − 0.631i)2-s + (0.528 + 1.64i)3-s + (1.20 + 1.59i)4-s + (2.66 − 1.54i)5-s + (0.372 − 2.42i)6-s + (−1.46 + 2.20i)7-s + (−0.511 − 2.78i)8-s + (−2.44 + 1.74i)9-s + (−4.34 + 0.264i)10-s + (−0.621 + 1.07i)11-s + (−2.00 + 2.82i)12-s + 5.98·13-s + (3.24 − 1.86i)14-s + (3.95 + 3.58i)15-s + (−1.10 + 3.84i)16-s + (−0.595 + 1.03i)17-s + ⋯
L(s)  = 1  + (−0.894 − 0.446i)2-s + (0.305 + 0.952i)3-s + (0.601 + 0.799i)4-s + (1.19 − 0.688i)5-s + (0.152 − 0.988i)6-s + (−0.552 + 0.833i)7-s + (−0.180 − 0.983i)8-s + (−0.813 + 0.581i)9-s + (−1.37 + 0.0835i)10-s + (−0.187 + 0.324i)11-s + (−0.577 + 0.816i)12-s + 1.66·13-s + (0.866 − 0.498i)14-s + (1.02 + 0.926i)15-s + (−0.277 + 0.960i)16-s + (−0.144 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.849 - 0.528i$
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: \((2,\ 168,\ (\ :1/2),\ 0.849 - 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.930409 + 0.265767i\)
\(L(\frac12)\) \(\approx\) \(0.930409 + 0.265767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.631i)T \)
3 \( 1 + (-0.528 - 1.64i)T \)
7 \( 1 + (1.46 - 2.20i)T \)
good5 \( 1 + (-2.66 + 1.54i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.621 - 1.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.98T + 13T^{2} \)
17 \( 1 + (0.595 - 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.614 + 1.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.56 + 1.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.19T + 29T^{2} \)
31 \( 1 + (-1.33 - 0.773i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.334 + 0.193i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.44T + 41T^{2} \)
43 \( 1 + 8.29iT - 43T^{2} \)
47 \( 1 + (3.34 + 5.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.25 + 9.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.22 + 1.86i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.7 - 6.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.21iT - 71T^{2} \)
73 \( 1 + (8.92 + 5.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.41 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.22iT - 83T^{2} \)
89 \( 1 + (-6.94 - 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−12.94800372766333154175729724913, −11.63885088831734133399559720505, −10.54258544731950476723070461494, −9.775854861911479327766395474476, −8.889153954834529624642533208361, −8.505969529882563175688937616572, −6.45261460706542591088995079548, −5.32411500777280023675178438952, −3.53779934452240452370399340784, −2.06288580055550487146749036414, 1.42247552979914511937626957468, 3.04595954920602485963164487899, 5.91692881480966728745152835451, 6.44240825633041821966262100308, 7.40939628229458030138601818048, 8.581532782999557114144851975032, 9.567720499508888225342437770589, 10.57726225495767187167270945693, 11.36231598464248258509898896724, 13.13397363629929166454289109585

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