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Properties

Label 1-687-687.161-r1-0-0
Degree $1$
Conductor $687$
Sign $0.726 - 0.687i$
Analytic cond. $73.8284$
Root an. cond. $73.8284$
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  + (−0.789 + 0.614i)2-s + (0.245 − 0.969i)4-s + (−0.546 + 0.837i)5-s + (−0.677 + 0.735i)7-s + (0.401 + 0.915i)8-s + (−0.0825 − 0.996i)10-s + (0.986 − 0.164i)11-s + (0.546 − 0.837i)13-s + (0.0825 − 0.996i)14-s + (−0.879 − 0.475i)16-s + (−0.546 + 0.837i)17-s + (0.546 + 0.837i)19-s + (0.677 + 0.735i)20-s + (−0.677 + 0.735i)22-s + (0.0825 − 0.996i)23-s + ⋯
L(s)  = 1  + (−0.789 + 0.614i)2-s + (0.245 − 0.969i)4-s + (−0.546 + 0.837i)5-s + (−0.677 + 0.735i)7-s + (0.401 + 0.915i)8-s + (−0.0825 − 0.996i)10-s + (0.986 − 0.164i)11-s + (0.546 − 0.837i)13-s + (0.0825 − 0.996i)14-s + (−0.879 − 0.475i)16-s + (−0.546 + 0.837i)17-s + (0.546 + 0.837i)19-s + (0.677 + 0.735i)20-s + (−0.677 + 0.735i)22-s + (0.0825 − 0.996i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 687 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 687 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(687\)    =    \(3 \cdot 229\)
Sign: $0.726 - 0.687i$
Analytic conductor: \(73.8284\)
Root analytic conductor: \(73.8284\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{687} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 687,\ (1:\ ),\ 0.726 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4240776934 - 0.1689171223i\)
\(L(\frac12)\) \(\approx\) \(0.4240776934 - 0.1689171223i\)
\(L(1)\) \(\approx\) \(0.5520633306 + 0.2204547961i\)
\(L(1)\) \(\approx\) \(0.5520633306 + 0.2204547961i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
229 \( 1 \)
good2 \( 1 + (-0.789 + 0.614i)T \)
5 \( 1 + (-0.546 + 0.837i)T \)
7 \( 1 + (-0.677 + 0.735i)T \)
11 \( 1 + (0.986 - 0.164i)T \)
13 \( 1 + (0.546 - 0.837i)T \)
17 \( 1 + (-0.546 + 0.837i)T \)
19 \( 1 + (0.546 + 0.837i)T \)
23 \( 1 + (0.0825 - 0.996i)T \)
29 \( 1 + (0.677 - 0.735i)T \)
31 \( 1 + (-0.986 + 0.164i)T \)
37 \( 1 + (-0.879 + 0.475i)T \)
41 \( 1 + (-0.789 + 0.614i)T \)
43 \( 1 + (-0.879 + 0.475i)T \)
47 \( 1 + (-0.789 - 0.614i)T \)
53 \( 1 + (-0.945 - 0.324i)T \)
59 \( 1 + (0.879 + 0.475i)T \)
61 \( 1 + (0.789 + 0.614i)T \)
67 \( 1 + (0.789 + 0.614i)T \)
71 \( 1 + (0.986 + 0.164i)T \)
73 \( 1 + (-0.401 - 0.915i)T \)
79 \( 1 + (-0.677 - 0.735i)T \)
83 \( 1 + (0.879 + 0.475i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.401 + 0.915i)T \)
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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

−22.43460822242033620886372773299, −21.68132643798078275338534155896, −20.60354837210227103858588785736, −20.01991550244862456875380141412, −19.55294812741535148114298160088, −18.70476928662559284479210735287, −17.57433076430713066835894705060, −16.973752042718012401323833352229, −16.09621131013332990326358707970, −15.729566136806238868218816586951, −14.02104893791591927123840430362, −13.29527631383981743665288859753, −12.416965972139166913412918033606, −11.55677445954090999211631618192, −11.031288855880252418666922864593, −9.62371855468923310837371171901, −9.24279628601276299648642581933, −8.40548335056601122281016194097, −7.13533309092621956080227359129, −6.79105183480998612130195539091, −4.9851590875705151347618160143, −3.92974781205008439730425102763, −3.355273775451131003250316578552, −1.74848716807932249923556558343, −0.86235910772599337888824967569, 0.18138449948242098221859835837, 1.62380449186098100244045762242, 2.91531065808693658827798164950, 3.89080566013395505174402540379, 5.417633273685017915986996554479, 6.42342545068547920686882658542, 6.72921701669272034442403960139, 8.13879942514318538759229518757, 8.55393790264122892218432856894, 9.748369006961436646923482611086, 10.422064658824240559825877286279, 11.36187481600292918359430567676, 12.17729835304129073160670252427, 13.40566688288686777516785599832, 14.5938183493762751979752286373, 15.01070401622661559552795537024, 15.88847731436917892612915825683, 16.514441381605878954586205001244, 17.62133449553210451290075512414, 18.35107397754099463774309378423, 19.01634559404058365239037541679, 19.65342356861963614465187991420, 20.431329756309786441484333455942, 21.86114386999909131284112538814, 22.59247245663869932818053090243

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