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Properties

Label 1-1337-1337.418-r1-0-0
Degree $1$
Conductor $1337$
Sign $0.666 - 0.745i$
Analytic cond. $143.680$
Root an. cond. $143.680$
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  + (0.137 − 0.990i)2-s + (0.298 + 0.954i)3-s + (−0.962 − 0.272i)4-s + (−0.716 + 0.697i)5-s + (0.986 − 0.164i)6-s + (−0.401 + 0.915i)8-s + (−0.821 + 0.569i)9-s + (0.592 + 0.805i)10-s + (−0.998 + 0.0550i)11-s + (−0.0275 − 0.999i)12-s + (0.677 + 0.735i)13-s + (−0.879 − 0.475i)15-s + (0.851 + 0.523i)16-s + (0.754 − 0.656i)17-s + (0.451 + 0.892i)18-s + (0.592 − 0.805i)19-s + ⋯
L(s)  = 1  + (0.137 − 0.990i)2-s + (0.298 + 0.954i)3-s + (−0.962 − 0.272i)4-s + (−0.716 + 0.697i)5-s + (0.986 − 0.164i)6-s + (−0.401 + 0.915i)8-s + (−0.821 + 0.569i)9-s + (0.592 + 0.805i)10-s + (−0.998 + 0.0550i)11-s + (−0.0275 − 0.999i)12-s + (0.677 + 0.735i)13-s + (−0.879 − 0.475i)15-s + (0.851 + 0.523i)16-s + (0.754 − 0.656i)17-s + (0.451 + 0.892i)18-s + (0.592 − 0.805i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1337\)    =    \(7 \cdot 191\)
Sign: $0.666 - 0.745i$
Analytic conductor: \(143.680\)
Root analytic conductor: \(143.680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1337} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1337,\ (1:\ ),\ 0.666 - 0.745i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8955506788 - 0.4006730717i\)
\(L(\frac12)\) \(\approx\) \(0.8955506788 - 0.4006730717i\)
\(L(1)\) \(\approx\) \(0.8306499138 - 0.04854671178i\)
\(L(1)\) \(\approx\) \(0.8306499138 - 0.04854671178i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.137 - 0.990i)T \)
3 \( 1 + (0.298 + 0.954i)T \)
5 \( 1 + (-0.716 + 0.697i)T \)
11 \( 1 + (-0.998 + 0.0550i)T \)
13 \( 1 + (0.677 + 0.735i)T \)
17 \( 1 + (0.754 - 0.656i)T \)
19 \( 1 + (0.592 - 0.805i)T \)
23 \( 1 + (-0.592 + 0.805i)T \)
29 \( 1 + (-0.879 - 0.475i)T \)
31 \( 1 + (-0.904 - 0.426i)T \)
37 \( 1 + (0.904 - 0.426i)T \)
41 \( 1 + (-0.789 + 0.614i)T \)
43 \( 1 + (-0.986 + 0.164i)T \)
47 \( 1 + (-0.451 + 0.892i)T \)
53 \( 1 + (-0.998 + 0.0550i)T \)
59 \( 1 + (-0.904 - 0.426i)T \)
61 \( 1 + (0.754 + 0.656i)T \)
67 \( 1 + (-0.191 + 0.981i)T \)
71 \( 1 + (0.789 - 0.614i)T \)
73 \( 1 + (-0.451 - 0.892i)T \)
79 \( 1 + (0.851 + 0.523i)T \)
83 \( 1 + (0.401 - 0.915i)T \)
89 \( 1 + (-0.350 - 0.936i)T \)
97 \( 1 + (0.0825 - 0.996i)T \)
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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

−20.51671058816159640535788803198, −20.225010372583207586454011279307, −18.96115095560119428738716320508, −18.490175316154473777317315869377, −17.917535984099869810528909840814, −16.78089882673176408972400872824, −16.38200736050265525461695311175, −15.414676413738400453303155285320, −14.80652859245346131291182428991, −13.91503438965758597413158310884, −13.07546125916710188740798848172, −12.653683343345245449925719198785, −11.97015168841314166292619624184, −10.72975752962555643751165446331, −9.61922424585912867076092830301, −8.53586029250710661844193361424, −8.07248870931670046056835940522, −7.66278165390293484176102528864, −6.63320941606558372064630826642, −5.64292698748263191745655281352, −5.17386086971230076700770597489, −3.76060090203755628270862609950, −3.24537123259654779293415955261, −1.6193303790398609360627571765, −0.584663694819375674517279329, 0.29678894514397399055644694583, 1.898030853422869607950998021454, 2.9510630359005169957570110494, 3.42316634508913611592411871742, 4.29681205804519095629798180643, 5.08642093233680468813372241244, 5.99326473561190913929703552617, 7.525692524712086246643363096759, 8.121185967844839388757537166010, 9.2412479835248507276644064242, 9.7677951732338840473706558627, 10.61497029022008840339421087462, 11.47409057740926729628327358826, 11.54881270196938901936157371182, 12.99120508241772401423369696349, 13.74420489769316685299633450954, 14.43200194469908916420171266180, 15.21124854943595623429418022325, 15.91164980895833482369508633335, 16.67065129646926084847618064999, 17.96342821159385664981448479394, 18.50145947986851614461848206189, 19.237632787956199696059163095000, 20.0353372604861221239933526098, 20.609385449639068782724302294955

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