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Properties

Label 1-287-287.173-r1-0-0
Degree $1$
Conductor $287$
Sign $0.670 - 0.742i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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)2-s + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + i·6-s − 8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.866 + 0.5i)12-s − i·13-s − i·15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + i·6-s − 8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.866 + 0.5i)12-s − i·13-s − i·15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.670 - 0.742i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.670 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.287084105 - 0.5721366620i\)
\(L(\frac12)\) \(\approx\) \(1.287084105 - 0.5721366620i\)
\(L(1)\) \(\approx\) \(1.192437933 + 0.3937531845i\)
\(L(1)\) \(\approx\) \(1.192437933 + 0.3937531845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + iT \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + iT \)
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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

−25.50571329407097145644591942019, −24.17267656499802316055161324521, −23.63002003159238743015096789933, −22.71161225565312007813434001074, −21.68209509946457304872712546111, −20.83984049945644915335257211193, −19.90790015023414571689883179219, −19.219896304733547555431792292930, −18.47352937752303039459304973076, −17.75576430416668876534176302234, −15.6606862424496296212149969246, −15.123115155521803556292551901244, −14.00586104193791461961052305620, −13.540149581235889118879296404463, −12.294379464888286104575205825338, −11.58768171835556049377963889435, −10.38634235086714301961955519649, −9.54598023920678097050539108589, −8.30380415372535280479888168203, −7.21855842236868300997973749608, −6.175107620676078281444802215820, −4.5179690743473938366106471110, −3.53977830479087405671510733945, −2.576839427958674483466310426188, −1.61534636201442361746090665677, 0.30592826144324119399893609093, 2.65255932858765930823134304429, 3.6710055189952968844735960375, 4.78110706315961079672376482800, 5.446573074308624457647930613588, 7.15205774039466920743807280687, 8.12410526413384801218494823741, 8.63646849229054166541115705550, 9.73064431000234600451170162842, 11.1562222608433799067735993245, 12.57828361250631075729253511209, 13.26888899134913616543201442123, 14.08461336886238753387883949579, 15.310134643582235398336524557733, 15.80757604084867221522542166833, 16.4520009413874562572214328296, 17.71642232663511626199116252885, 18.76045995509602905545400806563, 20.21932066825764336860797214997, 20.5034187829644323770640897785, 21.67955377463253517961043617854, 22.46605118795522766721073613178, 23.59269009957370221336332515900, 24.53648626965823049173344765339, 24.891053655150500871473472481815

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