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Properties

Label 2-168-168.101-c1-0-5
Degree $2$
Conductor $168$
Sign $-0.443 - 0.896i$
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 + (−1.69 + 0.366i)3-s + (1.20 + 1.59i)4-s + (−2.66 + 1.54i)5-s + (−2.37 − 0.604i)6-s + (−1.46 + 2.20i)7-s + (0.511 + 2.78i)8-s + (2.73 − 1.24i)9-s + (−4.34 + 0.264i)10-s + (0.621 − 1.07i)11-s + (−2.62 − 2.26i)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 + (−0.977 + 0.211i)3-s + (0.601 + 0.799i)4-s + (−1.19 + 0.688i)5-s + (−0.969 − 0.246i)6-s + (−0.552 + 0.833i)7-s + (0.180 + 0.983i)8-s + (0.910 − 0.414i)9-s + (−1.37 + 0.0835i)10-s + (0.187 − 0.324i)11-s + (−0.756 − 0.653i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.443 - 0.896i$
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: \((2,\ 168,\ (\ :1/2),\ -0.443 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603348 + 0.972112i\)
\(L(\frac12)\) \(\approx\) \(0.603348 + 0.972112i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 0.366i)T \)
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

−12.94522811663831308672819213309, −12.02584976136845952508904038910, −11.41418554666917212864825121836, −10.65809994185919465960128200936, −8.857009525191136049724694887243, −7.59269209024349731859503071603, −6.45354901608888192441339024842, −5.79303404117915853911852319679, −4.23439191540686102128218932434, −3.26102707385630507415081315782, 1.00905599201979661993226028240, 3.80681228782733170981325195499, 4.43897338832240604633388142858, 5.93215886714571118066612173784, 6.87622732268785573608822717597, 8.126304074720731508485898975932, 9.895099341995364885230130682161, 10.92032230535651300735214414159, 11.55975016547637150357212857422, 12.55484004340723346050653828620

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