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Properties

Label 1-285-285.218-r0-0-0
Degree $1$
Conductor $285$
Sign $-0.999 + 0.0141i$
Analytic cond. $1.32353$
Root an. cond. $1.32353$
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  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (0.5 − 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s + (−0.984 − 0.173i)22-s + (−0.642 − 0.766i)23-s + (0.5 + 0.866i)26-s + (0.342 − 0.939i)28-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s + (−0.984 + 0.173i)32-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (0.5 − 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s + (−0.984 − 0.173i)22-s + (−0.642 − 0.766i)23-s + (0.5 + 0.866i)26-s + (0.342 − 0.939i)28-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s + (−0.984 + 0.173i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.999 + 0.0141i$
Analytic conductor: \(1.32353\)
Root analytic conductor: \(1.32353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 285,\ (0:\ ),\ -0.999 + 0.0141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002740097723 - 0.3867409698i\)
\(L(\frac12)\) \(\approx\) \(0.002740097723 - 0.3867409698i\)
\(L(1)\) \(\approx\) \(0.5195746906 - 0.3144498518i\)
\(L(1)\) \(\approx\) \(0.5195746906 - 0.3144498518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.342 - 0.939i)T \)
53 \( 1 + (-0.642 - 0.766i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.342 + 0.939i)T \)
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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

−25.814527546073727400421424421778, −25.364364766175300733869577125929, −24.17690709361917764352207523504, −23.516723608320519271859618922480, −22.44752713489861855810946440249, −21.97490623300827132821582427653, −20.15342293763942264456823053580, −19.62324589910641184840208418618, −18.65045230924070140655955044988, −17.3724456625091181578139849468, −17.060109533461666254065091093108, −15.88527703759745093056683940299, −15.08064699739062261986618126341, −14.18698673157070631961648588265, −13.12795017884380419381109734505, −12.25526098461698402612442617005, −10.56812774121267364974627945955, −9.80021324960550192141470327267, −8.97491468797988669441274586408, −7.61112557134239443108242623622, −6.93911448659228000098267769276, −5.92739715468942214084799129601, −4.68872115546161530778325127366, −3.616016387707683546526450096286, −1.68695593320861209744542635032, 0.282640639302088837589347570116, 2.133242723016792786771468190760, 3.08860387376699605931930217675, 4.21823881192530741114830668613, 5.557624885067614501540371093311, 6.89025805335273880356957004222, 8.17334428991079969502326161735, 9.28365515123617623885440903646, 9.78278960722947876060631012875, 11.11442268627249974028433766271, 11.898243294499381715430298249632, 12.79905079662677702134904489579, 13.70729626413754538529417067813, 14.77302241596856445746886655016, 16.2525863800362967193718657798, 16.825400156690305608776563466220, 18.1017639422802914755928255391, 18.85013935500457135979728125555, 19.62422175908373318159043948942, 20.38775129145487120264115931773, 21.619369509745839178188605765031, 22.17563597377543911334563309156, 22.88646263982842287643947098741, 24.32096028675099132502592834664, 25.1637048189712244855222329312

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