Properties

Label 1-168-168.11-r0-0-0
Degree $1$
Conductor $168$
Sign $0.0633 + 0.997i$
Analytic cond. $0.780188$
Root an. cond. $0.780188$
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)5-s + (0.5 + 0.866i)11-s − 13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s − 41-s + 43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s − 13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s − 41-s + 43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(0.780188\)
Root analytic conductor: \(0.780188\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 168,\ (0:\ ),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6594850301 + 0.6189605914i\)
\(L(\frac12)\) \(\approx\) \(0.6594850301 + 0.6189605914i\)
\(L(1)\) \(\approx\) \(0.8716241994 + 0.2985236742i\)
\(L(1)\) \(\approx\) \(0.8716241994 + 0.2985236742i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−27.34700688259878567187517607700, −26.6648742311191016204928827575, −25.25687991290486817500973168041, −24.41108289003133558338003765706, −23.73744283523970808013198762198, −22.5057034702252131457620774040, −21.56075309977618843677903791891, −20.45644065794825396745867001994, −19.62169175512878376237783243282, −18.74499923549795703552503073219, −17.29426786032352734251009339887, −16.5650449225101578929055999661, −15.60799531913482570788285553162, −14.4107046560596876841045709776, −13.32346539877746665139187313868, −12.17272723825973815685141154873, −11.45747734168602885349992403133, −9.965549114177424954717760193479, −8.86357391466434323668214883211, −7.93965432516971138731017093209, −6.622226261300865042144629423383, −5.18374247721784007210610071364, −4.206002317234120291872075684085, −2.68398606859280669200205477895, −0.765022630274107827981426045658, 1.937719625148746198100849801715, 3.38454857982509925054733587356, 4.53012870822821549176745456523, 6.12790912704331308957196393807, 7.207284212889681243449308012524, 8.136951101287144091583342251706, 9.73271867576872175345243366376, 10.51475388578296421685025063306, 11.85011142498226725210133216560, 12.54863767233249745226305130989, 14.23104888695860575554762028034, 14.795566233704683528762688785545, 15.834602462728824579088342317790, 17.15392342383461899089905913584, 17.97419858102378620324152964180, 19.286038229553340021082027091990, 19.7075293042252042604721878241, 21.16846134479338321200707722561, 22.1275488244425774054433836315, 23.00521330147949642862584461298, 23.80220626256366309116986686475, 25.13525183256493216434491188577, 25.875418760781783189512583384, 27.022370118350250746329929231167, 27.60171774759677379615679387122

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