Properties

Label 2-168-168.107-c1-0-1
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: $\chi_{168} (107, \cdot )$
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

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

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