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Properties

Label 1-497-497.2-r0-0-0
Degree $1$
Conductor $497$
Sign $-0.512 + 0.858i$
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  + (0.420 + 0.907i)2-s + (−0.925 − 0.379i)3-s + (−0.646 + 0.762i)4-s + (0.913 + 0.406i)5-s + (−0.0448 − 0.998i)6-s + (−0.963 − 0.266i)8-s + (0.712 + 0.701i)9-s + (0.0149 + 0.999i)10-s + (0.998 − 0.0598i)11-s + (0.887 − 0.460i)12-s + (−0.550 + 0.834i)13-s + (−0.691 − 0.722i)15-s + (−0.163 − 0.986i)16-s + (−0.978 − 0.207i)17-s + (−0.337 + 0.941i)18-s + (0.971 + 0.237i)19-s + ⋯
L(s)  = 1  + (0.420 + 0.907i)2-s + (−0.925 − 0.379i)3-s + (−0.646 + 0.762i)4-s + (0.913 + 0.406i)5-s + (−0.0448 − 0.998i)6-s + (−0.963 − 0.266i)8-s + (0.712 + 0.701i)9-s + (0.0149 + 0.999i)10-s + (0.998 − 0.0598i)11-s + (0.887 − 0.460i)12-s + (−0.550 + 0.834i)13-s + (−0.691 − 0.722i)15-s + (−0.163 − 0.986i)16-s + (−0.978 − 0.207i)17-s + (−0.337 + 0.941i)18-s + (0.971 + 0.237i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 497 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 497 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(497\)    =    \(7 \cdot 71\)
Sign: $-0.512 + 0.858i$
Analytic conductor: \(2.30805\)
Root analytic conductor: \(2.30805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{497} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 497,\ (0:\ ),\ -0.512 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6461392347 + 1.137412894i\)
\(L(\frac12)\) \(\approx\) \(0.6461392347 + 1.137412894i\)
\(L(1)\) \(\approx\) \(0.8941144001 + 0.6224953139i\)
\(L(1)\) \(\approx\) \(0.8941144001 + 0.6224953139i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( 1 \)
good2 \( 1 + (0.420 + 0.907i)T \)
3 \( 1 + (-0.925 - 0.379i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (0.998 - 0.0598i)T \)
13 \( 1 + (-0.550 + 0.834i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.971 + 0.237i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (0.473 - 0.880i)T \)
31 \( 1 + (-0.163 + 0.986i)T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (0.753 + 0.657i)T \)
47 \( 1 + (-0.925 + 0.379i)T \)
53 \( 1 + (-0.337 + 0.941i)T \)
59 \( 1 + (-0.842 - 0.538i)T \)
61 \( 1 + (-0.873 - 0.486i)T \)
67 \( 1 + (-0.646 + 0.762i)T \)
73 \( 1 + (0.575 - 0.817i)T \)
79 \( 1 + (0.251 + 0.967i)T \)
83 \( 1 + (-0.0448 + 0.998i)T \)
89 \( 1 + (-0.646 - 0.762i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−22.97491734408947495308277876385, −22.32349373077858468235392386823, −21.88205554434383481220867388816, −20.97857679750392800724115238980, −20.227220679471749115586446026717, −19.3649933148589339580249225487, −18.01655336517651378329909388708, −17.63321793842387235270896553404, −16.81405568230095929681631005289, −15.58913511497544893694681234683, −14.684096755147200197296376710633, −13.66744687035719229049383944054, −12.78497211778808843935391649625, −12.133139098232302603853329757911, −11.14239570724873808043922368980, −10.41696062846964909400980309429, −9.47012668072208003672289231320, −8.96937804259416758557902248251, −6.98933404826406071698279418884, −5.93599265323198341057403526331, −5.21290358296071765066974904356, −4.42863627040799512732544583061, −3.23107357284387854347271745332, −1.85392228752477623977434066490, −0.77547482486133236526661680112, 1.40789871517640323299092787745, 2.86387374407480421868250978535, 4.41616166316289780242930091226, 5.137373276592993344084473627078, 6.331625560233531129294486959, 6.62373406814054984695041376428, 7.5544587597408605343613269785, 9.02645996101422670553977891814, 9.73182571887979335465254030690, 11.07858604908136939414274952085, 11.935085336350065882455873562818, 12.829020472551397721225763682933, 13.807318246123643261843717011226, 14.288202101513943575027253655073, 15.461947791007235825580142375008, 16.47703755620866775163705888192, 17.124090131122833607685881030184, 17.73849290800576667929990677606, 18.48942500490160730519350160695, 19.47851107385226298478576000596, 21.081741567536342994550104711334, 21.77848183273490781907389224654, 22.47884305712533090434524866192, 22.9372371177193557594039251304, 24.156694937230038309139913432843

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