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Properties

Label 1-285-285.197-r0-0-0
Degree $1$
Conductor $285$
Sign $-0.334 + 0.942i$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·7-s + i·8-s − 11-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)22-s + (0.866 − 0.5i)23-s − 26-s + (−0.866 + 0.5i)28-s + (−0.5 − 0.866i)29-s + 31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·7-s + i·8-s − 11-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)22-s + (0.866 − 0.5i)23-s − 26-s + (−0.866 + 0.5i)28-s + (−0.5 − 0.866i)29-s + 31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.334 + 0.942i$
Analytic conductor: \(1.32353\)
Root analytic conductor: \(1.32353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 285,\ (0:\ ),\ -0.334 + 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056734054 + 1.495850082i\)
\(L(\frac12)\) \(\approx\) \(1.056734054 + 1.495850082i\)
\(L(1)\) \(\approx\) \(1.327489684 + 0.8275042864i\)
\(L(1)\) \(\approx\) \(1.327489684 + 0.8275042864i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 - T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (-0.866 - 0.5i)T \)
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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

−25.03899045764845333958416445748, −24.16807497696738675506874605325, −23.212546847215489634679983408020, −22.81738958104661634442474446411, −21.53658163173292839691586766584, −20.84504877484077191767035513743, −20.012754257575697776131660445842, −19.20663178310803033545961747704, −18.10923111338818532300212766975, −16.89814060272608874469507153140, −15.91950185921777644536457631644, −14.90002131567289736002652124831, −14.03639846742142146065079483847, −13.140096249188870414690925863281, −12.39170144915789583656484046710, −11.173935604081978579015279550172, −10.381791762053013710104294197405, −9.57863100179352774305238593837, −7.74935547778623501086899196536, −7.00234202203947553021440102202, −5.52717811176620141001145061034, −4.784323250038106332555725428992, −3.51254442669015492040088512145, −2.548474567597972662953249905672, −0.94042955075183080453130378443, 2.21115914753080501166492233369, 3.075124755642830723526642942299, 4.57714149026488683439879124109, 5.41322022880367365955909553582, 6.36574292799655142019498541766, 7.59036717419258338689375611648, 8.423098497483924792371889384846, 9.71828871430969957895635318974, 11.07956603311719560422375604948, 12.15056679335097071058651327758, 12.735459757415680726379542869011, 13.85105816392652082534289963066, 14.89269530075390776291833298994, 15.44978327907503845497880620479, 16.50284119551473303562231984893, 17.3592826531327757159688303323, 18.5490696694362966504677289812, 19.43270345004749253976535147763, 20.998134775370547955637070645693, 21.21545040655368754139041756343, 22.34865580684619349239687880704, 23.06900880248776503236404645031, 24.17773804118585513210833592788, 24.663654719162465637257790892229, 25.760745183506564881097972474607

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