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Properties

Label 1-1617-1617.530-r1-0-0
Degree $1$
Conductor $1617$
Sign $-0.139 - 0.990i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (0.525 − 0.850i)2-s + (−0.447 − 0.894i)4-s + (0.193 − 0.981i)5-s + (−0.995 − 0.0896i)8-s + (−0.733 − 0.680i)10-s + (0.473 + 0.880i)13-s + (−0.599 + 0.800i)16-s + (0.842 + 0.538i)17-s + (−0.978 + 0.207i)19-s + (−0.963 + 0.266i)20-s + (−0.365 + 0.930i)23-s + (−0.925 − 0.379i)25-s + (0.998 + 0.0598i)26-s + (0.936 + 0.351i)29-s + (−0.913 + 0.406i)31-s + (0.365 + 0.930i)32-s + ⋯
L(s)  = 1  + (0.525 − 0.850i)2-s + (−0.447 − 0.894i)4-s + (0.193 − 0.981i)5-s + (−0.995 − 0.0896i)8-s + (−0.733 − 0.680i)10-s + (0.473 + 0.880i)13-s + (−0.599 + 0.800i)16-s + (0.842 + 0.538i)17-s + (−0.978 + 0.207i)19-s + (−0.963 + 0.266i)20-s + (−0.365 + 0.930i)23-s + (−0.925 − 0.379i)25-s + (0.998 + 0.0598i)26-s + (0.936 + 0.351i)29-s + (−0.913 + 0.406i)31-s + (0.365 + 0.930i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (530, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ -0.139 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.780002791 - 2.047456225i\)
\(L(\frac12)\) \(\approx\) \(1.780002791 - 2.047456225i\)
\(L(1)\) \(\approx\) \(1.122817460 - 0.7839826216i\)
\(L(1)\) \(\approx\) \(1.122817460 - 0.7839826216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.525 - 0.850i)T \)
5 \( 1 + (0.193 - 0.981i)T \)
13 \( 1 + (0.473 + 0.880i)T \)
17 \( 1 + (0.842 + 0.538i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (0.936 + 0.351i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.163 - 0.986i)T \)
41 \( 1 + (0.995 + 0.0896i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.971 + 0.237i)T \)
53 \( 1 + (-0.887 - 0.460i)T \)
59 \( 1 + (0.420 - 0.907i)T \)
61 \( 1 + (0.887 - 0.460i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.963 + 0.266i)T \)
73 \( 1 + (0.971 - 0.237i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (-0.473 + 0.880i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−20.77606173676807553611117564557, −19.68699623935286767779388301522, −18.62782224588829455446515977963, −18.22337834775655731611691546130, −17.422442604604356297209387409854, −16.6993655571068462528150412473, −15.80735258883048376421180504286, −15.20261665316309750178099575700, −14.46538055210551029144561637028, −13.95651483233282405641245518740, −13.06206932572401479244536391197, −12.39329855701603222854331284467, −11.43157711466076520753576882182, −10.57618018069788171833588400512, −9.799432036102013073433966309482, −8.73966234121070066056072274819, −7.93080952879358203417669123809, −7.27916428761487133944312509758, −6.31254936838409097023951968904, −5.91959823684921704896763616851, −4.86553812803845766136392868398, −3.926634329186429344596880445, −3.06521590351430130539953792710, −2.36894795618567286175738359140, −0.64389580220857201836995395250, 0.64634791617823662006066828106, 1.57569484067597983165912820066, 2.211083195305989281502792192968, 3.6397106699849476559379060860, 4.08456348814683348008108148900, 5.10341715545803581995528924830, 5.74995489875301559951442060388, 6.58505860265045994736627418132, 7.92221923931228115008061241780, 8.839447268471287147238217531498, 9.35264941884471551193496510754, 10.265997086288677212538901023, 11.01163287767131379014333317092, 11.90785423818975572454070410791, 12.543858259446946214544853162730, 13.064206188375305536709699758863, 14.0616322273664023597801647719, 14.42203119044742205377355960184, 15.62364533558802370600381955457, 16.23159958009376591824964448850, 17.19641426492756509120285057660, 17.85794764109231838283361571934, 18.919098319179033573947316058261, 19.38066994084932406889573141269, 20.16231091718123321556411684243

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