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Properties

Label 1-585-585.193-r0-0-0
Degree $1$
Conductor $585$
Sign $0.182 - 0.983i$
Analytic cond. $2.71672$
Root an. cond. $2.71672$
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 + (−0.5 + 0.866i)4-s − 7-s + 8-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)22-s − i·23-s + (0.5 − 0.866i)28-s + (0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 7-s + 8-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)22-s − i·23-s + (0.5 − 0.866i)28-s + (0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.182 - 0.983i$
Analytic conductor: \(2.71672\)
Root analytic conductor: \(2.71672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 585,\ (0:\ ),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5661788927 - 0.4709283885i\)
\(L(\frac12)\) \(\approx\) \(0.5661788927 - 0.4709283885i\)
\(L(1)\) \(\approx\) \(0.6409457742 - 0.2581575791i\)
\(L(1)\) \(\approx\) \(0.6409457742 - 0.2581575791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + iT \)
43 \( 1 - iT \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + T \)
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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

−23.467056749861752953709616442425, −22.8466654708595191757063306313, −21.798136230366118009296171049422, −20.93567180372862571897631903888, −19.51080529347165852500040420384, −19.250649719589067018689554512517, −18.35224769121309936106404678653, −17.34464274706374968025726688934, −16.69152667448995912205377787218, −15.72595326152947983357841124852, −15.35091632738218797503068088920, −14.10827122357070753532043156074, −13.363913928234883644236302008895, −12.55689976271202983317360334600, −11.14759849039640316723934942056, −10.18752573980977310689031657727, −9.59836766424743264517103244954, −8.47728890014292476582282920684, −7.794137352359963104359563368, −6.6742963140600970427066413575, −5.98216560419942575621472706122, −5.051963935941762803834418640956, −3.78762995071055094017467182613, −2.54284251736629672153938476877, −0.89602453239525300769408758032, 0.611445897462519046187327277635, 2.18718231252543687126297524006, 2.98067842561189165686432023668, 4.01818527923200329063907618140, 5.10613905023919480117599950308, 6.443610225028693951067569038048, 7.501435762353327587028294150316, 8.38374524565711116771357155370, 9.3999785928861855820619243576, 10.181293363619109938627915068773, 10.73524555595481973900050786094, 12.086711670835802000682382967086, 12.59073953400105605720652930334, 13.36616096871976476699969937992, 14.40432431346325164889442394709, 15.65207266165976095085174429731, 16.48819699995294779817004001964, 17.16605667918274809819653885565, 18.42131560290049767815174291519, 18.68977684549916968734067821200, 19.71758144597759244955179112232, 20.43434201049425499820546345049, 21.20313822477874737607925487310, 22.01458801405226096717856351785, 23.01149795355382647556405447881

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