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Properties

Label 1-497-497.17-r0-0-0
Degree $1$
Conductor $497$
Sign $0.809 + 0.586i$
Analytic cond. $2.30805$
Root an. cond. $2.30805$
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  + (−0.978 − 0.207i)2-s + (−0.669 + 0.743i)3-s + (0.913 + 0.406i)4-s + (−0.913 + 0.406i)5-s + (0.809 − 0.587i)6-s + (−0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.978 − 0.207i)10-s + (−0.669 + 0.743i)11-s + (−0.913 + 0.406i)12-s + (0.309 − 0.951i)13-s + (0.309 − 0.951i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.104 + 0.994i)18-s + (0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.669 + 0.743i)3-s + (0.913 + 0.406i)4-s + (−0.913 + 0.406i)5-s + (0.809 − 0.587i)6-s + (−0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.978 − 0.207i)10-s + (−0.669 + 0.743i)11-s + (−0.913 + 0.406i)12-s + (0.309 − 0.951i)13-s + (0.309 − 0.951i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.104 + 0.994i)18-s + (0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 497 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 497 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(497\)    =    \(7 \cdot 71\)
Sign: $0.809 + 0.586i$
Analytic conductor: \(2.30805\)
Root analytic conductor: \(2.30805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{497} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 497,\ (0:\ ),\ 0.809 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4340695617 + 0.1407282077i\)
\(L(\frac12)\) \(\approx\) \(0.4340695617 + 0.1407282077i\)
\(L(1)\) \(\approx\) \(0.4648804949 + 0.08779717099i\)
\(L(1)\) \(\approx\) \(0.4648804949 + 0.08779717099i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (-0.913 + 0.406i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + T \)
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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

−23.73216825501404870309184226908, −23.21569325074812650232690430340, −21.91058308305252141716414158500, −20.84513477248374195386227660648, −19.81484526003173987557124449079, −19.216689257208750740372972443539, −18.47854752271763221710510477018, −17.755660219660895206265261002122, −16.7471363947672544246135178580, −16.084539044701396746210044250059, −15.539914473571081966562119719938, −14.06125374513523372191270825395, −13.0737832704931824451306942157, −11.95173578744435442526766018174, −11.31124315222905190492439766491, −10.763853272047822628669277602671, −9.23906390407445485993609379850, −8.48947031132189839632849005568, −7.5040588541500475238666686416, −6.95065168891490827259623828146, −5.79417873502446712083561270784, −4.85170272740123433975166807126, −3.187654363655360574995973669691, −1.78847267161408176701096350466, −0.65670896169812508460637378546, 0.68163852250886630281796845020, 2.54883820164480415145328336134, 3.55142053204838480600246745276, 4.63318701375261763708647309546, 5.9344925852642297455905597711, 6.99495909754885049240266920913, 7.85371588775859090635849691261, 8.82155466676634799997348773859, 9.96857322360470345122112369642, 10.567379314796250307817359028578, 11.33293300240148211676214047839, 12.0771438836686272645345667422, 13.0369966684538050867337044518, 14.95255025167579716286702619313, 15.40755675841333562248276786576, 16.03915480654104441670674803033, 17.01134032806024176822885839368, 17.9100752395833506156683938661, 18.43802283572222135962651427378, 19.530385869855170678476666904661, 20.58815344389915021870183780471, 20.735561480664927842697750369866, 22.356502539560397243261489426805, 22.5913671552293683103776648073, 23.757560948010111920215582475195

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