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Properties

Label 2-168-168.107-c1-0-25
Degree $2$
Conductor $168$
Sign $-0.606 + 0.795i$
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 + (1.28 − 1.16i)3-s + (−1.87 + 0.691i)4-s + (0.646 − 1.11i)5-s + (−1.93 − 1.49i)6-s + (−2.42 − 1.05i)7-s + (1.42 + 2.44i)8-s + (0.300 − 2.98i)9-s + (−1.71 − 0.621i)10-s + (1.60 − 0.923i)11-s + (−1.60 + 3.06i)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 + (0.741 − 0.670i)3-s + (−0.938 + 0.345i)4-s + (0.289 − 0.500i)5-s + (−0.790 − 0.612i)6-s + (−0.917 − 0.397i)7-s + (0.505 + 0.862i)8-s + (0.100 − 0.994i)9-s + (−0.543 − 0.196i)10-s + (0.482 − 0.278i)11-s + (−0.463 + 0.885i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.606 + 0.795i$
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: \((2,\ 168,\ (\ :1/2),\ -0.606 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.521570 - 1.05387i\)
\(L(\frac12)\) \(\approx\) \(0.521570 - 1.05387i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 + 1.16i)T \)
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

−12.41298483115470484342892566551, −11.78259999597698606233054245526, −10.20277873122971903652348692818, −9.447495680836040207012848944724, −8.626031334092287052219808452786, −7.46698274030228592419573816939, −6.05213369059129726560955154253, −4.14508390519355216517318699201, −2.99020430464332434279028621672, −1.30541863121714637983470714322, 2.94918722738597742170919614001, 4.37038990552655184531632166253, 5.80935168518252970442871598988, 6.84362933811811248296482808250, 8.093600763973893493308767789537, 9.066295386522726771970838023230, 9.885621488640161939052095086000, 10.62595833942705129626235972325, 12.53735172029307551584106949160, 13.41634803096224418682213801667

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