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Properties

Label 2-168-168.125-c3-0-31
Degree $2$
Conductor $168$
Sign $0.220 - 0.975i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.60 − 1.09i)2-s + (4.05 + 3.24i)3-s + (5.61 + 5.69i)4-s + 13.2i·5-s + (−7.03 − 12.9i)6-s + (18.4 − 1.29i)7-s + (−8.43 − 20.9i)8-s + (5.90 + 26.3i)9-s + (14.4 − 34.5i)10-s + 49.5·11-s + (4.28 + 41.3i)12-s − 28.7·13-s + (−49.6 − 16.7i)14-s + (−42.9 + 53.7i)15-s + (−0.896 + 63.9i)16-s − 69.2·17-s + ⋯
L(s)  = 1  + (−0.922 − 0.385i)2-s + (0.780 + 0.624i)3-s + (0.702 + 0.712i)4-s + 1.18i·5-s + (−0.478 − 0.877i)6-s + (0.997 − 0.0700i)7-s + (−0.372 − 0.927i)8-s + (0.218 + 0.975i)9-s + (0.457 − 1.09i)10-s + 1.35·11-s + (0.103 + 0.994i)12-s − 0.614·13-s + (−0.947 − 0.320i)14-s + (−0.740 + 0.924i)15-s + (−0.0140 + 0.999i)16-s − 0.988·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ 0.220 - 0.975i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.25051 + 0.998886i\)
\(L(\frac12)\) \(\approx\) \(1.25051 + 0.998886i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.60 + 1.09i)T \)
3 \( 1 + (-4.05 - 3.24i)T \)
7 \( 1 + (-18.4 + 1.29i)T \)
good5 \( 1 - 13.2iT - 125T^{2} \)
11 \( 1 - 49.5T + 1.33e3T^{2} \)
13 \( 1 + 28.7T + 2.19e3T^{2} \)
17 \( 1 + 69.2T + 4.91e3T^{2} \)
19 \( 1 - 24.7T + 6.85e3T^{2} \)
23 \( 1 + 85.5iT - 1.21e4T^{2} \)
29 \( 1 + 111.T + 2.43e4T^{2} \)
31 \( 1 - 236. iT - 2.97e4T^{2} \)
37 \( 1 - 261. iT - 5.06e4T^{2} \)
41 \( 1 - 471.T + 6.89e4T^{2} \)
43 \( 1 + 261. iT - 7.95e4T^{2} \)
47 \( 1 + 217.T + 1.03e5T^{2} \)
53 \( 1 + 11.8T + 1.48e5T^{2} \)
59 \( 1 + 236. iT - 2.05e5T^{2} \)
61 \( 1 + 754.T + 2.26e5T^{2} \)
67 \( 1 - 163. iT - 3.00e5T^{2} \)
71 \( 1 + 478. iT - 3.57e5T^{2} \)
73 \( 1 + 1.03e3iT - 3.89e5T^{2} \)
79 \( 1 + 148.T + 4.93e5T^{2} \)
83 \( 1 - 739. iT - 5.71e5T^{2} \)
89 \( 1 - 1.34e3T + 7.04e5T^{2} \)
97 \( 1 - 183. iT - 9.12e5T^{2} \)
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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

−12.16676297582221350588376604804, −11.10475884105211626010810628495, −10.60515599919751184585574997352, −9.469319209901490561105129559162, −8.662867866773481001135893805229, −7.54903534869228811313531470644, −6.65382522682234895426522944977, −4.42038555020251333644799233445, −3.12576367504354393921901642581, −1.89153882202007935278511666791, 0.985829040925127567039549751118, 2.04283469624498903978965512258, 4.38089830829879989000180147904, 5.89455473715061096986072469385, 7.27128730017936732859120808493, 8.050303520936717992426221661312, 9.138383228851650584637705464220, 9.348510580196324740913050206739, 11.24746523340074607172041629027, 11.98996576248213341982294114690

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