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Properties

Label 1-573-573.50-r1-0-0
Degree $1$
Conductor $573$
Sign $-0.341 + 0.940i$
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  + (0.574 − 0.818i)2-s + (−0.340 − 0.940i)4-s + (−0.789 + 0.614i)5-s + (−0.809 − 0.587i)7-s + (−0.965 − 0.261i)8-s + (0.0495 + 0.998i)10-s + (0.879 − 0.475i)11-s + (0.746 − 0.665i)13-s + (−0.945 + 0.324i)14-s + (−0.768 + 0.639i)16-s + (0.148 − 0.988i)17-s + (0.0495 − 0.998i)19-s + (0.846 + 0.533i)20-s + (0.115 − 0.993i)22-s + (0.934 + 0.355i)23-s + ⋯
L(s)  = 1  + (0.574 − 0.818i)2-s + (−0.340 − 0.940i)4-s + (−0.789 + 0.614i)5-s + (−0.809 − 0.587i)7-s + (−0.965 − 0.261i)8-s + (0.0495 + 0.998i)10-s + (0.879 − 0.475i)11-s + (0.746 − 0.665i)13-s + (−0.945 + 0.324i)14-s + (−0.768 + 0.639i)16-s + (0.148 − 0.988i)17-s + (0.0495 − 0.998i)19-s + (0.846 + 0.533i)20-s + (0.115 − 0.993i)22-s + (0.934 + 0.355i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 573 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.341 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 573 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.341 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(573\)    =    \(3 \cdot 191\)
Sign: $-0.341 + 0.940i$
Analytic conductor: \(61.5774\)
Root analytic conductor: \(61.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{573} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 573,\ (1:\ ),\ -0.341 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4056392240 - 0.5787190508i\)
\(L(\frac12)\) \(\approx\) \(-0.4056392240 - 0.5787190508i\)
\(L(1)\) \(\approx\) \(0.7622188085 - 0.6232139871i\)
\(L(1)\) \(\approx\) \(0.7622188085 - 0.6232139871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.574 - 0.818i)T \)
5 \( 1 + (-0.789 + 0.614i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.879 - 0.475i)T \)
13 \( 1 + (0.746 - 0.665i)T \)
17 \( 1 + (0.148 - 0.988i)T \)
19 \( 1 + (0.0495 - 0.998i)T \)
23 \( 1 + (0.934 + 0.355i)T \)
29 \( 1 + (-0.997 - 0.0660i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.677 + 0.735i)T \)
41 \( 1 + (-0.945 - 0.324i)T \)
43 \( 1 + (0.922 + 0.386i)T \)
47 \( 1 + (-0.180 - 0.983i)T \)
53 \( 1 + (-0.431 + 0.901i)T \)
59 \( 1 + (-0.980 - 0.197i)T \)
61 \( 1 + (0.701 + 0.712i)T \)
67 \( 1 + (-0.461 + 0.887i)T \)
71 \( 1 + (-0.601 + 0.799i)T \)
73 \( 1 + (-0.724 + 0.689i)T \)
79 \( 1 + (0.997 - 0.0660i)T \)
83 \( 1 + (0.934 - 0.355i)T \)
89 \( 1 + (0.973 + 0.229i)T \)
97 \( 1 + (-0.909 - 0.416i)T \)
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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

−23.57496023021693249540550397979, −22.881822602156450445177818408071, −22.2384621514643692391961408535, −21.21833784190133823796547397132, −20.47009803368469200299076254949, −19.31558076469860857526672851600, −18.713259718901537468627929952893, −17.44445106650098485877206131272, −16.53120072577230624559967566769, −16.1702234194361009755631321265, −15.13858237571348458756847190684, −14.58079158885016040166285325816, −13.37782814854322179511759575821, −12.457002191281444713550616294864, −12.17303331123556552735970835932, −10.97015433996406749949332122506, −9.2965500701368130959682181143, −8.86281342437006378155446745612, −7.85989353200970922813794914162, −6.81895238034713950243874120505, −6.07061780482391200495011271221, −5.03567841687788984824660710687, −3.88618149719220779685759697123, −3.45826294922854551978968818699, −1.64800729821752829653671901523, 0.16146420717257798148248809174, 1.08977851814639282363931117466, 2.834805432905857881780768513288, 3.43639787252681089286141680175, 4.20114874220051219085022281623, 5.49026154627738041498981290252, 6.57747175322715810641239991503, 7.31196293444409014198586102997, 8.80729543700611776071003255610, 9.62498315236295646997858299230, 10.73498280585640501312836701428, 11.2558780937268486055395139374, 12.065094543031142717668459816542, 13.22279440354111198314284854577, 13.68874362576095468534090272030, 14.788520886564661599562817776149, 15.50409424877779144232670882998, 16.39018649723669511419051468635, 17.59887561229316397952114525712, 18.7915967695852487702826784252, 19.10335195493782000798208453934, 20.14195780797579834014708167463, 20.49855094240257530402327538391, 21.916615861885624364826910520342, 22.43329198629829777967736938081

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