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Properties

Label 1-287-287.3-r0-0-0
Degree $1$
Conductor $287$
Sign $0.951 + 0.308i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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 + (0.258 − 0.965i)3-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (0.707 − 0.707i)6-s + i·8-s + (−0.866 − 0.5i)9-s + (0.5 + 0.866i)10-s + (0.258 − 0.965i)11-s + (0.965 − 0.258i)12-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)15-s + (−0.5 + 0.866i)16-s + (−0.965 − 0.258i)17-s + (−0.5 − 0.866i)18-s + (0.258 + 0.965i)19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.258 − 0.965i)3-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (0.707 − 0.707i)6-s + i·8-s + (−0.866 − 0.5i)9-s + (0.5 + 0.866i)10-s + (0.258 − 0.965i)11-s + (0.965 − 0.258i)12-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)15-s + (−0.5 + 0.866i)16-s + (−0.965 − 0.258i)17-s + (−0.5 − 0.866i)18-s + (0.258 + 0.965i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.951 + 0.308i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.951 + 0.308i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.449682161 + 0.3870720630i\)
\(L(\frac12)\) \(\approx\) \(2.449682161 + 0.3870720630i\)
\(L(1)\) \(\approx\) \(1.971420025 + 0.2302924684i\)
\(L(1)\) \(\approx\) \(1.971420025 + 0.2302924684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (-0.965 - 0.258i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.258 + 0.965i)T \)
53 \( 1 + (0.258 - 0.965i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.965 + 0.258i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−25.368964637028244067359203950955, −24.67274807014510361447480548297, −23.428268787930381432724931797663, −22.52423187743005670902531723930, −21.75662115041477288399694455241, −21.115399241246169675973553855839, −20.090125615412183923791475187440, −19.88844180052838652543774834219, −18.10168758566449839869055712135, −17.19665391016633807109366465154, −15.92965409692708575209514469825, −15.28652318649459206899468693528, −14.32663106316360909733029759192, −13.38173919170204386971570403447, −12.68814143537215528978047508920, −11.27933331937658036266988675246, −10.52512590088626314274532401678, −9.53232740519454853379013454237, −8.83788167872870049899422938228, −6.98015149811779121747157565694, −5.65081731900621364933103147217, −4.97163921701299214036105883087, −3.959410228832759351366390754774, −2.777252570876516704182574751490, −1.602102821355657011217906199780, 1.73499082484724130846918554607, 2.76334956935394027821815585506, 3.88285884177867420222347680600, 5.589426885121781491611623657256, 6.30877953783808385507927847706, 6.99699366456349682895434261078, 8.25031191892256351565052343940, 9.14590632967559051337836024063, 10.94021009160506772335611297375, 11.69756431806106006258534666449, 12.934735908327434776749043295009, 13.66646823692550364802366594133, 14.15536618812249953755707628307, 15.1234212770309902011831138023, 16.49810341838325377645173502800, 17.21555978677768845019957584954, 18.340877250432182077741205598294, 18.978108964962152364529320292652, 20.443321840647875385812388819548, 21.0922358066066184540269825201, 22.262224747908020064536632922347, 22.80386799524743432744626182092, 24.06913960441320109985701674638, 24.48917757460672969339836596204, 25.40331402802011914217279976470

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