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Properties

Label 1-143-143.86-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.845 + 0.533i$
Analytic cond. $15.3674$
Root an. cond. $15.3674$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.845 + 0.533i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.845 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6155584995 - 2.131033006i\)
\(L(\frac12)\) \(\approx\) \(-0.6155584995 - 2.131033006i\)
\(L(1)\) \(\approx\) \(0.6740374231 - 1.292382383i\)
\(L(1)\) \(\approx\) \(0.6740374231 - 1.292382383i\)

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

−28.250535908572071651749564949232, −27.27196270152238160859492709650, −26.61193567260109346518496570412, −25.79332849902366332397516269302, −24.71614125848503665773188431794, −23.611410016341170492364640266485, −22.67968816633952917954147717973, −21.78084374123217504962032692679, −21.077214582174314043997322215436, −19.84515211895258548622145823568, −18.39134054383488499705528077122, −17.36180758287139759592277733573, −16.10502859267229471613543409177, −15.35582271742413322624139449171, −14.53011483280584240558199769183, −13.89771369798775393401834680943, −12.054931756409029026734655367127, −11.20738889294284890816053245641, −9.82239525955031510910537178591, −8.31152325210771935825530607160, −7.71264389213651107462423010562, −6.09012356922166876440921833013, −4.88886617706873683157852623129, −3.856302775190035468433199555689, −2.73991565024286285051246544937, 0.71289791187042106037142959570, 1.69376199372161707017828630959, 3.27579969279247181760545058492, 4.61174288118068786333298108762, 5.75251510698644735160350740595, 7.478268960985959374728224593504, 8.42132364792460108168531139082, 9.742791855500837313675342647701, 11.43797884537703043540901356343, 11.942669285938330138651787826855, 12.984356606917503485053038269964, 13.984997159294117483302321460344, 14.76547691750694888704154879860, 16.28075818748141277598100663225, 17.78303133561749116888082409029, 18.599379285297258648244271119715, 19.78661288107637769320255193322, 20.35762262183415292301597474290, 21.1813955783576215876978178553, 22.730267957333828447768460914372, 23.63112764994511965165490393068, 24.23262724708031437600319776420, 25.045539797762868658204220372705, 26.7563032209134144726438141675, 27.75183045099431639560201073046

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