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Properties

Label 2-168-168.149-c0-0-1
Degree $2$
Conductor $168$
Sign $0.895 + 0.444i$
Analytic cond. $0.0838429$
Root an. cond. $0.289556$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s − 0.999·14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s − 0.999·14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(0.0838429\)
Root analytic conductor: \(0.289556\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :0),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7950034699\)
\(L(\frac12)\) \(\approx\) \(0.7950034699\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−13.16302320037969944480696072567, −11.74342602875900181326235820481, −10.67912812258416350923297324413, −10.44604020380389069179961353883, −9.240445997654200465171850406238, −7.983189307617049929617508331280, −6.45840513587427204111383799514, −4.90885335948892201490429824040, −3.63772716762778935296786994848, −2.94693018909009501599165498144, 2.75321242507591566263979590084, 4.41569849154479348210501520321, 5.72205468344983056887725200940, 6.85306887210919170952066197257, 7.943921129814027549837558082507, 8.642314643704530090554462014690, 9.606095076322175874873032937294, 11.83714033401652427485785498122, 12.45124170982219093015462280986, 13.00038542653060252709731992770

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