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Properties

Label 1-473-473.379-r0-0-0
Degree $1$
Conductor $473$
Sign $0.596 - 0.802i$
Analytic cond. $2.19660$
Root an. cond. $2.19660$
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  + (0.983 − 0.178i)2-s + (−0.691 − 0.722i)3-s + (0.936 − 0.351i)4-s + (−0.393 + 0.919i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)7-s + (0.858 − 0.512i)8-s + (−0.0448 + 0.998i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)12-s + (0.753 + 0.657i)13-s + (0.134 − 0.990i)14-s + (0.936 − 0.351i)15-s + (0.753 − 0.657i)16-s + (0.753 − 0.657i)17-s + (0.134 + 0.990i)18-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)2-s + (−0.691 − 0.722i)3-s + (0.936 − 0.351i)4-s + (−0.393 + 0.919i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)7-s + (0.858 − 0.512i)8-s + (−0.0448 + 0.998i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)12-s + (0.753 + 0.657i)13-s + (0.134 − 0.990i)14-s + (0.936 − 0.351i)15-s + (0.753 − 0.657i)16-s + (0.753 − 0.657i)17-s + (0.134 + 0.990i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $0.596 - 0.802i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ 0.596 - 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.844746266 - 0.9275287702i\)
\(L(\frac12)\) \(\approx\) \(1.844746266 - 0.9275287702i\)
\(L(1)\) \(\approx\) \(1.524821250 - 0.4663040324i\)
\(L(1)\) \(\approx\) \(1.524821250 - 0.4663040324i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.983 - 0.178i)T \)
3 \( 1 + (-0.691 - 0.722i)T \)
5 \( 1 + (-0.393 + 0.919i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.753 + 0.657i)T \)
17 \( 1 + (0.753 - 0.657i)T \)
19 \( 1 + (-0.550 + 0.834i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.691 + 0.722i)T \)
31 \( 1 + (0.983 - 0.178i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.134 - 0.990i)T \)
47 \( 1 + (-0.550 + 0.834i)T \)
53 \( 1 + (-0.393 - 0.919i)T \)
59 \( 1 + (0.858 + 0.512i)T \)
61 \( 1 + (0.983 + 0.178i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.0448 - 0.998i)T \)
73 \( 1 + (-0.995 - 0.0896i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.983 + 0.178i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (-0.0448 + 0.998i)T \)
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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

−23.61319494719324306695376166529, −23.28326906539414030815380473889, −22.230687744623163789503150332394, −21.384769614144779262958145971385, −20.9832300694365295249250592341, −20.09110306113911062350434079276, −18.96278075663572688362454563112, −17.53274395292768932310454528055, −16.94056517795297324496964095416, −15.92422572271637772177853017425, −15.38840991566893070015176090463, −14.80986267751288709650940325603, −13.266666181751645743546850075864, −12.6483606150769429200653973290, −11.6632232665011933289658261916, −11.254468199916982664871455038690, −9.97217387393177697457394081661, −8.73250723231028635529519662800, −7.94025686744419352129757349738, −6.402100759557524499791263906565, −5.57755237415341322098472217303, −4.958116199848212862583594115061, −4.02685526502605552915323649350, −3.01036943098134535997591065036, −1.33700322206283863813384707770, 1.09642324365074935673225072987, 2.29100674098078336044159151430, 3.56523912656921286732284753055, 4.44982659848703518624944823293, 5.650859742673537216803708786784, 6.632890546096778639123906985413, 7.1696347273308840754406400988, 8.075775876000042854743473472290, 10.12889529002909100670536723614, 10.9053688251161012640735928881, 11.41672131006726926312393652496, 12.34085313878374441798357989097, 13.28450554926617379334308362507, 14.1680357740110493054200074223, 14.63282586550665610116208521874, 16.08982871352459509657470212922, 16.59912599273493368357298813701, 17.77449438987425723013277019251, 18.87474712700523682515885091383, 19.232719811930498623011306921290, 20.52446455022430169047031625163, 21.20319183590091083949677975438, 22.43146478361872105743133634882, 22.92498517640134910483276507276, 23.4905620608241975135644325869

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