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Properties

Label 1-143-143.138-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.348 + 0.937i$
Analytic cond. $0.664089$
Root an. cond. $0.664089$
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  + (0.587 + 0.809i)2-s + (0.309 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (0.587 − 0.809i)5-s + (−0.587 + 0.809i)6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s + 10-s − 12-s + (0.309 + 0.951i)14-s + (0.951 + 0.309i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.951 − 0.309i)18-s + (0.951 − 0.309i)19-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.309 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (0.587 − 0.809i)5-s + (−0.587 + 0.809i)6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s + 10-s − 12-s + (0.309 + 0.951i)14-s + (0.951 + 0.309i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.951 − 0.309i)18-s + (0.951 − 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.348 + 0.937i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.348 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9707735065 + 1.397141237i\)
\(L(\frac12)\) \(\approx\) \(0.9707735065 + 1.397141237i\)
\(L(1)\) \(\approx\) \(1.211249958 + 1.000581543i\)
\(L(1)\) \(\approx\) \(1.211249958 + 1.000581543i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.951 - 0.309i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.951 - 0.309i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−28.40547952153029459457411580318, −27.01971910655450572753097022404, −26.05237132756430142704168773295, −24.67750424620730997032057605811, −24.06313708158533464909247695119, −22.9761216730911839280807237621, −22.057069316517838310348501158324, −20.95655161645955884032585582110, −20.07944579754882302579464263185, −19.045400368848677166407012966808, −18.08038898639763807737741972330, −17.526130860781960619018343191245, −15.29229156478985210444341016112, −14.14171146076412207790501741421, −13.885569274262829200710811617153, −12.616639969396615596789595383694, −11.497879020873651665684601758633, −10.64794222417523569145493299472, −9.32044241226579780145065545245, −7.8559904260276820480673465979, −6.55208641733410707258323049604, −5.48964594953182135326530027883, −3.80607662232747198743761026819, −2.425052668006634599057543512098, −1.526433299841239666634110166307, 2.3741348056110019924672793090, 4.05687238167291273771312599671, 5.02081012811860514606590643250, 5.73939400227801243467517037452, 7.59467284679671539670238384346, 8.73656626957322425289591602990, 9.404786837964049563888248807877, 11.1086020342673518268081369309, 12.3070318196124142876885585143, 13.711544620212885428007398085978, 14.29523032567562811613737852294, 15.51877265973721679747465705608, 16.21304681506273876116001795406, 17.29527535590355061233750605636, 18.0874886491568709925610534642, 20.2251154253549143851084175186, 20.78903333610676334833025672972, 21.79451572081709430717732298388, 22.397608518034659337717718694208, 23.98899999297826503680984217832, 24.59679872412464469190635375910, 25.56034772603579049350994609220, 26.484354633940719946201454957093, 27.48162474831397996914745661313, 28.30535994283989517963746873723

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