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Properties

Label 1-143-143.16-r0-0-0
Degree $1$
Conductor $143$
Sign $0.859 + 0.511i$
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  + (−0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.809 − 0.587i)5-s + (−0.104 + 0.994i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (0.669 + 0.743i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (0.309 − 0.951i)18-s + (0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.809 − 0.587i)5-s + (−0.104 + 0.994i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (0.669 + 0.743i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (0.309 − 0.951i)18-s + (0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2587703625 + 0.07114493359i\)
\(L(\frac12)\) \(\approx\) \(0.2587703625 + 0.07114493359i\)
\(L(1)\) \(\approx\) \(0.4279717183 - 0.1782032251i\)
\(L(1)\) \(\approx\) \(0.4279717183 - 0.1782032251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−28.03708814823213852470892447132, −26.90867612166904870462689182608, −26.58197577721344128771165739530, −25.23456976048851388125639611243, −24.12971063743230035358452061791, −23.0327609342316433905856912407, −22.764437411022794032743764645671, −21.84245598686042080081810013232, −20.043090766793186766163322637323, −18.70049040038358762635473943224, −18.20977982286363463288242220209, −16.76819608302030824323676037111, −16.135681327836425186364378414, −15.395468545010337315277525488482, −14.14091293342419285607700005287, −12.81361395694597620536977972807, −11.728222413314219462996038665551, −10.37182252965247907668967439186, −9.50821266135433758716762845092, −7.81623798236309761467131639081, −6.8423733968291819662071386720, −6.02765269704266549298318823283, −4.62311645829543717503541262095, −3.502263033950399751216364276507, −0.30916625773036625190798221159, 1.33746408528306888309668065736, 3.298104953219341733370646114649, 4.499251712685502426008770464559, 5.68188209097952762884138196519, 7.25866652151817248578530318111, 8.66586635825252738314201659982, 9.84555526598036174977836823871, 10.9581439315872796173654040826, 11.965341652627698447889703549233, 12.61695668130955219835292013176, 13.50081372062490247765059949102, 15.44982422480608566936598069002, 16.4372265397927585257669118099, 17.41708601528916237483711050980, 18.53706548904144125759617654219, 19.45295571988445882953567654614, 20.16685133221474820191635342030, 21.69988017287169101459085928694, 22.24098942522632055930266503089, 23.4195883120064597487495563076, 23.90383835408558518456770513987, 25.558370917961655092538851396322, 26.87506004107282375137686761283, 27.703132965492331389523660922837, 28.60472789330015981698922399129

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