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Properties

Label 1-573-573.218-r1-0-0
Degree $1$
Conductor $573$
Sign $-0.627 + 0.778i$
Analytic cond. $61.5774$
Root an. cond. $61.5774$
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  + (0.846 + 0.533i)2-s + (0.431 + 0.901i)4-s + (0.879 + 0.475i)5-s + (0.309 + 0.951i)7-s + (−0.115 + 0.993i)8-s + (0.490 + 0.871i)10-s + (0.401 + 0.915i)11-s + (0.652 + 0.757i)13-s + (−0.245 + 0.969i)14-s + (−0.627 + 0.778i)16-s + (0.999 − 0.0330i)17-s + (0.490 − 0.871i)19-s + (−0.0495 + 0.998i)20-s + (−0.148 + 0.988i)22-s + (0.909 − 0.416i)23-s + ⋯
L(s)  = 1  + (0.846 + 0.533i)2-s + (0.431 + 0.901i)4-s + (0.879 + 0.475i)5-s + (0.309 + 0.951i)7-s + (−0.115 + 0.993i)8-s + (0.490 + 0.871i)10-s + (0.401 + 0.915i)11-s + (0.652 + 0.757i)13-s + (−0.245 + 0.969i)14-s + (−0.627 + 0.778i)16-s + (0.999 − 0.0330i)17-s + (0.490 − 0.871i)19-s + (−0.0495 + 0.998i)20-s + (−0.148 + 0.988i)22-s + (0.909 − 0.416i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 573 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 573 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(573\)    =    \(3 \cdot 191\)
Sign: $-0.627 + 0.778i$
Analytic conductor: \(61.5774\)
Root analytic conductor: \(61.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{573} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 573,\ (1:\ ),\ -0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.172365948 + 4.537896029i\)
\(L(\frac12)\) \(\approx\) \(2.172365948 + 4.537896029i\)
\(L(1)\) \(\approx\) \(1.823272816 + 1.444227508i\)
\(L(1)\) \(\approx\) \(1.823272816 + 1.444227508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.846 + 0.533i)T \)
5 \( 1 + (0.879 + 0.475i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.401 + 0.915i)T \)
13 \( 1 + (0.652 + 0.757i)T \)
17 \( 1 + (0.999 - 0.0330i)T \)
19 \( 1 + (0.490 - 0.871i)T \)
23 \( 1 + (0.909 - 0.416i)T \)
29 \( 1 + (0.934 - 0.355i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.986 + 0.164i)T \)
41 \( 1 + (-0.245 - 0.969i)T \)
43 \( 1 + (-0.574 - 0.818i)T \)
47 \( 1 + (0.213 - 0.976i)T \)
53 \( 1 + (0.995 - 0.0990i)T \)
59 \( 1 + (0.461 - 0.887i)T \)
61 \( 1 + (-0.340 + 0.940i)T \)
67 \( 1 + (-0.277 - 0.960i)T \)
71 \( 1 + (-0.371 + 0.928i)T \)
73 \( 1 + (0.863 - 0.504i)T \)
79 \( 1 + (-0.934 - 0.355i)T \)
83 \( 1 + (0.909 + 0.416i)T \)
89 \( 1 + (0.956 + 0.293i)T \)
97 \( 1 + (0.701 - 0.712i)T \)
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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

−22.82319671211178313389844940262, −21.69037163088473404972804991332, −21.11555341938907968722902688455, −20.46953082031983177726719127490, −19.7108671920723780800109080191, −18.680723208489560046055037926153, −17.75733154899352831865267787842, −16.667772697454982637266165063091, −16.13209415769732679725717593185, −14.71667742681372168821317599552, −14.06777417274883978133990989932, −13.4316838121005304004847700982, −12.68948001570687078942892346366, −11.652197829475826375467495278148, −10.66610047107582190082751644255, −10.12205169979867127841066070554, −9.03608612961752538272640708333, −7.8498716819370991970715467226, −6.58586765515469838753599467166, −5.68224524114287774160527081917, −5.02461857346691732936616406306, −3.74519018615350264648083561061, −3.04420781631052232182559355768, −1.35930377489000594401649258874, −1.027451221354082981540400412426, 1.69040538587508854640156623251, 2.54700939067569472888701136151, 3.62498722753215007350245532101, 4.949091872023150112352700575410, 5.55360701454400805965058854104, 6.62482166693734541928956393193, 7.18873852931163880833792122838, 8.60271810002314931400029857108, 9.30311553259433785227253493445, 10.55612273076856673841274844600, 11.66196580219105859497648700588, 12.27823274391551712472775927952, 13.34394377424266190149162515295, 14.09127614725313500428363293168, 14.84375344066195110814733513571, 15.477000717935862671973601071493, 16.57172801033038729632018876332, 17.4015387039181142877298341994, 18.14595600981770492008186782820, 18.99293323576072695157556722068, 20.403806090163748221830810400068, 21.14507924385637192641086900815, 21.7286229574834657485384523878, 22.48517258283212078411310788054, 23.22527384800275204455202889425

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