Properties

Label 1-280-280.67-r0-0-0
Degree $1$
Conductor $280$
Sign $0.999 - 0.0333i$
Analytic cond. $1.30031$
Root an. cond. $1.30031$
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)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s i·13-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)23-s + i·27-s + 29-s + (0.5 + 0.866i)31-s + (0.866 + 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + 41-s i·43-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s i·13-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)23-s + i·27-s + 29-s + (0.5 + 0.866i)31-s + (0.866 + 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + 41-s i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0333i$
Analytic conductor: \(1.30031\)
Root analytic conductor: \(1.30031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 280,\ (0:\ ),\ 0.999 - 0.0333i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9056956267 + 0.01512635008i\)
\(L(\frac12)\) \(\approx\) \(0.9056956267 + 0.01512635008i\)
\(L(1)\) \(\approx\) \(0.8421968431 + 0.04511952603i\)
\(L(1)\) \(\approx\) \(0.8421968431 + 0.04511952603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - iT \)
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

−25.37916858690717815669916241289, −24.771896595574849001135429210168, −23.56565809898251496051028671274, −23.0713457496231216503449613296, −22.21070685028351255514029549834, −21.16298606172786333046880531830, −20.13793111461551787575495023227, −19.102502477305461489370669435627, −18.081991520338193445942768912573, −17.58692283047981425045744485266, −16.4862892498665274622577476753, −15.62971205161765521790827899510, −14.49196846526787872622968052688, −13.253213041321129619827655361935, −12.47242638224329929152987304190, −11.72268911433610104396918196595, −10.430525810109869636837641745812, −9.88260430723860392807905028181, −8.01601179014521219994223269845, −7.497285820051615223988839534491, −6.08033727989539212553514472336, −5.39814211242563379859592432392, −4.12336686886065819204820742573, −2.49587398027428853189327406215, −1.10398270488693037058004677493, 0.88869719583456950788523523570, 2.80448570502580121670879938084, 4.13492534224405015441688736722, 5.15213719568540117388191531781, 6.126620653925983041245902901691, 7.15982848555647717450619192928, 8.55889776133390155440468662493, 9.65170084422078113181500951934, 10.56246974623367299712337846115, 11.53957778314267913182334279650, 12.22332026461603908869947700577, 13.57272715140847985316974109961, 14.47255124113908834119280011158, 15.90894253134805311282731050972, 16.19860609285451312282518165919, 17.2897777994744258999924396105, 18.2421324061940764498113298514, 19.04807061284631746268086073270, 20.33842653144171260057897868738, 21.38552194537092507803626584128, 21.82539318259992666132333784962, 22.948132366537596994898805853999, 23.74377230482364539657856359513, 24.42941976117095080598383761217, 25.81546672136811417033014990014

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