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Properties

Label 2-168-168.11-c1-0-27
Degree $2$
Conductor $168$
Sign $-0.879 - 0.475i$
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  + (0.248 − 1.39i)2-s + (−1.64 − 0.531i)3-s + (−1.87 − 0.691i)4-s + (−0.646 − 1.11i)5-s + (−1.14 + 2.16i)6-s + (−2.42 + 1.05i)7-s + (−1.42 + 2.44i)8-s + (2.43 + 1.75i)9-s + (−1.71 + 0.621i)10-s + (−1.60 − 0.923i)11-s + (2.72 + 2.13i)12-s − 2.25i·13-s + (0.862 + 3.64i)14-s + (0.470 + 2.18i)15-s + (3.04 + 2.59i)16-s + (−3.89 − 2.24i)17-s + ⋯
L(s)  = 1  + (0.175 − 0.984i)2-s + (−0.951 − 0.306i)3-s + (−0.938 − 0.345i)4-s + (−0.289 − 0.500i)5-s + (−0.469 + 0.883i)6-s + (−0.917 + 0.397i)7-s + (−0.505 + 0.862i)8-s + (0.811 + 0.584i)9-s + (−0.543 + 0.196i)10-s + (−0.482 − 0.278i)11-s + (0.786 + 0.617i)12-s − 0.625i·13-s + (0.230 + 0.973i)14-s + (0.121 + 0.565i)15-s + (0.760 + 0.648i)16-s + (−0.944 − 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.879 - 0.475i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0982725 + 0.388203i\)
\(L(\frac12)\) \(\approx\) \(0.0982725 + 0.388203i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.248 + 1.39i)T \)
3 \( 1 + (1.64 + 0.531i)T \)
7 \( 1 + (2.42 - 1.05i)T \)
good5 \( 1 + (0.646 + 1.11i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.60 + 0.923i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.25iT - 13T^{2} \)
17 \( 1 + (3.89 + 2.24i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.80 + 4.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.519 - 0.900i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.32T + 29T^{2} \)
31 \( 1 + (-3.69 - 2.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.18 + 4.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.39iT - 41T^{2} \)
43 \( 1 + 6.02T + 43T^{2} \)
47 \( 1 + (5.90 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.02 - 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.57 - 5.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.65 - 4.41i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.05 + 5.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (4.38 - 7.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.37 - 1.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.74iT - 83T^{2} \)
89 \( 1 + (-8.31 + 4.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.73T + 97T^{2} \)
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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

−12.23456972348416349873068801163, −11.32423951604989180362821353116, −10.49960689166485962972974380800, −9.420873809649680975923662322830, −8.321146753444575451317294328158, −6.66697376052375133375587455035, −5.45921869468200133511657284401, −4.44164623139874484561473699599, −2.66155589498659644139504503803, −0.38988360354091037854047983105, 3.68389346623579403853276658222, 4.73673647787965965399968416541, 6.25676147603936468314737771289, 6.69693660853566589731066639165, 7.947517467907656076006501708080, 9.442379061643272968049162810537, 10.26658687518914382102884695734, 11.36758742431661754057461596456, 12.66234621890203116692508776619, 13.20581076885583522422269985791

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