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Properties

Label 2-168-168.5-c1-0-26
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: $\chi_{168} (5, \cdot )$
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

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

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