Properties

Label 2-168-168.5-c1-0-4
Degree $2$
Conductor $168$
Sign $-0.164 - 0.986i$
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  + (0.968 + 1.03i)2-s + (−1.70 − 0.283i)3-s + (−0.124 + 1.99i)4-s + (2.24 + 1.29i)5-s + (−1.36 − 2.03i)6-s + (−2.53 + 0.751i)7-s + (−2.17 + 1.80i)8-s + (2.83 + 0.967i)9-s + (0.839 + 3.57i)10-s + (1.63 + 2.83i)11-s + (0.777 − 3.37i)12-s + 0.912·13-s + (−3.23 − 1.88i)14-s + (−3.47 − 2.85i)15-s + (−3.96 − 0.495i)16-s + (−2.39 − 4.14i)17-s + ⋯
L(s)  = 1  + (0.684 + 0.728i)2-s + (−0.986 − 0.163i)3-s + (−0.0621 + 0.998i)4-s + (1.00 + 0.580i)5-s + (−0.556 − 0.830i)6-s + (−0.958 + 0.284i)7-s + (−0.769 + 0.638i)8-s + (0.946 + 0.322i)9-s + (0.265 + 1.13i)10-s + (0.493 + 0.855i)11-s + (0.224 − 0.974i)12-s + 0.253·13-s + (−0.863 − 0.504i)14-s + (−0.897 − 0.737i)15-s + (−0.992 − 0.123i)16-s + (−0.580 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.164 - 0.986i$
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: \((2,\ 168,\ (\ :1/2),\ -0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823868 + 0.972954i\)
\(L(\frac12)\) \(\approx\) \(0.823868 + 0.972954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.968 - 1.03i)T \)
3 \( 1 + (1.70 + 0.283i)T \)
7 \( 1 + (2.53 - 0.751i)T \)
good5 \( 1 + (-2.24 - 1.29i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.63 - 2.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.912T + 13T^{2} \)
17 \( 1 + (2.39 + 4.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.66 + 4.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.45 - 2.57i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.35T + 29T^{2} \)
31 \( 1 + (-8.18 + 4.72i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.59 - 0.922i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.91T + 41T^{2} \)
43 \( 1 + 8.00iT - 43T^{2} \)
47 \( 1 + (3.29 - 5.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.841 - 1.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.50 + 0.867i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.72 - 8.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.8 - 6.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.603iT - 71T^{2} \)
73 \( 1 + (1.29 - 0.746i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0625 - 0.108i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.246iT - 83T^{2} \)
89 \( 1 + (1.80 - 3.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.10iT - 97T^{2} \)
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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

−13.37751341470917737118302621687, −12.18629497415561900478030430591, −11.39803678788912379616057847506, −9.989915889463087959364669546761, −9.170154356663552360628043998960, −7.15068681839608164871523519090, −6.66013260119380336898291779928, −5.73545372366248196485033723739, −4.60722069855076863050943159039, −2.72957468578452998867637310210, 1.26943677318261768463919832395, 3.47595805236681908010016286582, 4.83996526974132224532912493833, 6.03904620973082650554095283400, 6.45960226776180441299589367305, 8.892838504627941309285121940382, 9.917204921250097277124331954268, 10.52575654519775239942115260504, 11.62958767132022746889325209448, 12.61602512789472027606559692641

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