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Properties

Label 1-143-143.98-r0-0-0
Degree $1$
Conductor $143$
Sign $0.0257 + 0.999i$
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  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + i·5-s + (0.866 + 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s − 12-s + 14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − i·18-s + (0.866 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + i·5-s + (0.866 + 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s − 12-s + 14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − i·18-s + (0.866 + 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0257 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0257 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.0257 + 0.999i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.0257 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2889488098 + 0.2964985948i\)
\(L(\frac12)\) \(\approx\) \(0.2889488098 + 0.2964985948i\)
\(L(1)\) \(\approx\) \(0.5008886454 + 0.1268363814i\)
\(L(1)\) \(\approx\) \(0.5008886454 + 0.1268363814i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + iT \)
7 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 + T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−28.2511403817126314934554038385, −27.20005927321229592026808855621, −26.39427653045273366725477321300, −25.30681654409985872683810112721, −24.34758567984641413512488190249, −22.72186817046662120523934182082, −21.97164907503055197303491498796, −20.83217691958936888527391006876, −20.25829328982433202898403950374, −19.09836151573842953095471254039, −17.8611513421152623735078586940, −16.908504023059659189786282705272, −16.07545197254423839538757772047, −15.48413118878754345473421450230, −13.36456184217038050491918396918, −12.21466660451499753153601929018, −11.497143651629879380626935774348, −10.11074717478164107611675019357, −9.32004423720021525683676425631, −8.59545641129922321305767347052, −6.84544615220317604662355899820, −5.41640807540703984081121232744, −4.065673427411477802622228083811, −2.66927412301754200105765668114, −0.524098281670507298995813375147, 1.51752741336828903508777699201, 3.12988208258577825893234458674, 5.503617661483251237598407065901, 6.67796865105198330824618797838, 7.09572891366971251026043789084, 8.36350954639182555041902002879, 9.95358167301379851124959033173, 10.74471010814774866571088257689, 11.82884111816652185518463980596, 13.327100541014025314977608925438, 14.32102045615135898002913103914, 15.5991926569412771844840691744, 16.65648484452825800790319185435, 17.62052779329595995208755878944, 18.41110987208063597769578178493, 19.322261687571690865944540756807, 19.94057961407214203307715080275, 21.89379158221126870381589311390, 23.02666650547530976957401196597, 23.54937989377774721816283286859, 24.80311655978768964533288997984, 25.67108473988350187257424629854, 26.44671720986417811049385584810, 27.44652866755151293382522273102, 28.74994189664554799944157858502

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