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Properties

Label 1-287-287.97-r0-0-0
Degree $1$
Conductor $287$
Sign $0.735 + 0.677i$
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  + (0.951 + 0.309i)2-s + (−0.707 + 0.707i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (−0.891 + 0.453i)6-s + (0.587 + 0.809i)8-s − i·9-s + (0.809 − 0.587i)10-s + (0.156 − 0.987i)11-s + (−0.987 + 0.156i)12-s + (0.453 + 0.891i)13-s + (0.156 + 0.987i)15-s + (0.309 + 0.951i)16-s + (0.987 + 0.156i)17-s + (0.309 − 0.951i)18-s + (0.453 − 0.891i)19-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.707 + 0.707i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (−0.891 + 0.453i)6-s + (0.587 + 0.809i)8-s − i·9-s + (0.809 − 0.587i)10-s + (0.156 − 0.987i)11-s + (−0.987 + 0.156i)12-s + (0.453 + 0.891i)13-s + (0.156 + 0.987i)15-s + (0.309 + 0.951i)16-s + (0.987 + 0.156i)17-s + (0.309 − 0.951i)18-s + (0.453 − 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.735 + 0.677i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896061039 + 0.7408873882i\)
\(L(\frac12)\) \(\approx\) \(1.896061039 + 0.7408873882i\)
\(L(1)\) \(\approx\) \(1.604514687 + 0.4611703222i\)
\(L(1)\) \(\approx\) \(1.604514687 + 0.4611703222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (0.156 - 0.987i)T \)
13 \( 1 + (0.453 + 0.891i)T \)
17 \( 1 + (0.987 + 0.156i)T \)
19 \( 1 + (0.453 - 0.891i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.987 + 0.156i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (0.987 - 0.156i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.951 - 0.309i)T \)
67 \( 1 + (-0.156 - 0.987i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.891 - 0.453i)T \)
97 \( 1 + (-0.156 - 0.987i)T \)
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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

−25.20887894450231417321851606304, −24.44191524790911687212266944360, −23.28847763915966883799134427512, −22.632699495530593403214797411475, −22.28459453591168692655889155077, −20.97109894133945516456428133919, −20.20717924896453690138312950256, −18.85792516198424283241130635748, −18.346009879189654873953530522825, −17.24880442082161176596607156862, −16.20336617402930722313280050227, −14.94271679661192000562908755364, −14.22561357130667946080626515526, −13.19999190196585879477512012205, −12.46005762538750451550825576060, −11.56476279956285018211140535622, −10.51879945470220663656190400963, −9.920499046231610946656590190037, −7.73672233512389606512585975409, −6.92949649984513405201920417956, −5.90350952192729743659166463847, −5.29640088115105790969671772988, −3.73535581650225925292210675681, −2.448137921814464857417287751379, −1.444032410090491335038593477039, 1.46610208172458995482604136996, 3.315175228943662799822110157054, 4.25274630176230580010579100653, 5.445264429291843209251909854692, 5.8146115495942141967271989884, 7.07767089941079544698500047622, 8.62905310767481631151337929634, 9.5569324955538457756298162583, 10.93620210126037886094337914825, 11.65958003284997390249580844688, 12.6265107512919325669301719323, 13.65820113453647599343774997617, 14.438549218793434653312972248339, 15.75456807170440109465774670626, 16.37179316712699458752520544820, 16.96283951585443028581170698696, 17.98379877152171727449137674759, 19.56400947243385396130768316111, 20.731220566127709997821443022205, 21.39319787560375177860527488198, 21.83920913523847191469682193949, 22.92610438734042623907819079436, 23.95226558891823388833132643243, 24.272447301046868841935902638932, 25.65275260315312416658261732307

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