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Properties

Label 2-168-168.53-c0-0-0
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: $\chi_{168} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :0),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4604818675\)
\(L(\frac12)\) \(\approx\) \(0.4604818675\)
\(L(1)\) 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

−12.84377716228207501419088051201, −11.69990351447646209675072534419, −11.12604064038424515327874362899, −10.09861140063608070135486276296, −9.423081223041522445327077865868, −8.468093894064484209807210098068, −6.63905656850873214618772627775, −5.54820606904850479034972179275, −3.82044615430027564516087224396, −2.69249934410288796452239146739, 1.42722757428129667973181858411, 4.55193127382454352826726131071, 5.70653117352097134240690750479, 6.79207488065491108055610177215, 7.53416320918595835474679745506, 8.799447406525906404167765232095, 9.765355133904787142195595799780, 10.77864463473598852248893201810, 12.22043201297493245743789487034, 13.15706231648850386518619222628

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