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Properties

Label 1-1547-1547.506-r0-0-0
Degree $1$
Conductor $1547$
Sign $-0.564 - 0.825i$
Analytic cond. $7.18423$
Root an. cond. $7.18423$
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 + (−0.866 + 0.5i)5-s + i·6-s + 8-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)10-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)12-s + 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s − i·20-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + i·6-s + 8-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)10-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)12-s + 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s − i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1547\)    =    \(7 \cdot 13 \cdot 17\)
Sign: $-0.564 - 0.825i$
Analytic conductor: \(7.18423\)
Root analytic conductor: \(7.18423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1547} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1547,\ (0:\ ),\ -0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1437534610 - 0.2723688723i\)
\(L(\frac12)\) \(\approx\) \(0.1437534610 - 0.2723688723i\)
\(L(1)\) \(\approx\) \(0.3985751463 - 0.1748968883i\)
\(L(1)\) \(\approx\) \(0.3985751463 - 0.1748968883i\)

Euler product

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

−20.678708143117522936603471747083, −20.12778594421170538196252970440, −18.946322966001027606017080155191, −18.5594745239183212717424692091, −17.57236166638797313712656360849, −16.98587322016441186522506436876, −16.341561240520615423521200247339, −15.50402611131665748985823081134, −15.39874135377722661975126930985, −14.318486864201059503128055878084, −13.22457142841098308956258027581, −12.35144381160405628746538459404, −11.70585306894577562754554491826, −10.56857228210504511915838659719, −10.20665130837353861116416758296, −9.25283394429259711127472982686, −8.315817043013363702377056161890, −7.71707157724506956240583736023, −6.81227535404071922679904890933, −5.92719945125071564716781511460, −5.18509307816103033938535853889, −4.42858744781941038062590691388, −3.783915067829660737420099002561, −1.99018233305958504855946329080, −0.637859946540043574412336806185, 0.27326851671928299625919885716, 1.435181213139187800104554956147, 2.56818120873024571092483941468, 3.33180199982441820222262112454, 4.44215299884003799023108563874, 5.15561150915502292665985537872, 6.42867600279381995159862971324, 7.20092236696757113953886722644, 7.98826484609156416801572642045, 8.5557667800457875442311512194, 9.86427188970310701185790956693, 10.69666060490181901759054107308, 11.02935437156490435500576976920, 11.893995615775567263473864473809, 12.37729405694632906973196880591, 13.30829802027911008798509849461, 13.89037030987167086176824200381, 15.25916866436700993969724410441, 16.03489740078952637707112604117, 16.65098918563872014926714918712, 17.62697560657772521026314318352, 18.19097171736253259948914636030, 18.74668711856899096311399738615, 19.50387993102706066717668985377, 19.95582039347575295629643560825

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