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Properties

Label 1-573-573.59-r1-0-0
Degree $1$
Conductor $573$
Sign $0.224 - 0.974i$
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  + (−0.894 + 0.446i)2-s + (0.601 − 0.799i)4-s + (−0.945 − 0.324i)5-s + (−0.809 − 0.587i)7-s + (−0.180 + 0.983i)8-s + (0.991 − 0.131i)10-s + (−0.245 + 0.969i)11-s + (−0.627 + 0.778i)13-s + (0.986 + 0.164i)14-s + (−0.277 − 0.960i)16-s + (−0.922 − 0.386i)17-s + (0.991 + 0.131i)19-s + (−0.828 + 0.560i)20-s + (−0.213 − 0.976i)22-s + (−0.431 + 0.901i)23-s + ⋯
L(s)  = 1  + (−0.894 + 0.446i)2-s + (0.601 − 0.799i)4-s + (−0.945 − 0.324i)5-s + (−0.809 − 0.587i)7-s + (−0.180 + 0.983i)8-s + (0.991 − 0.131i)10-s + (−0.245 + 0.969i)11-s + (−0.627 + 0.778i)13-s + (0.986 + 0.164i)14-s + (−0.277 − 0.960i)16-s + (−0.922 − 0.386i)17-s + (0.991 + 0.131i)19-s + (−0.828 + 0.560i)20-s + (−0.213 − 0.976i)22-s + (−0.431 + 0.901i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 573 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 573 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(573\)    =    \(3 \cdot 191\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(61.5774\)
Root analytic conductor: \(61.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{573} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 573,\ (1:\ ),\ 0.224 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1703801360 - 0.1356066492i\)
\(L(\frac12)\) \(\approx\) \(0.1703801360 - 0.1356066492i\)
\(L(1)\) \(\approx\) \(0.4515959247 + 0.08326480787i\)
\(L(1)\) \(\approx\) \(0.4515959247 + 0.08326480787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.894 + 0.446i)T \)
5 \( 1 + (-0.945 - 0.324i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.245 + 0.969i)T \)
13 \( 1 + (-0.627 + 0.778i)T \)
17 \( 1 + (-0.922 - 0.386i)T \)
19 \( 1 + (0.991 + 0.131i)T \)
23 \( 1 + (-0.431 + 0.901i)T \)
29 \( 1 + (0.340 + 0.940i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (-0.401 + 0.915i)T \)
41 \( 1 + (0.986 - 0.164i)T \)
43 \( 1 + (0.490 + 0.871i)T \)
47 \( 1 + (0.846 + 0.533i)T \)
53 \( 1 + (-0.371 - 0.928i)T \)
59 \( 1 + (-0.863 - 0.504i)T \)
61 \( 1 + (-0.518 + 0.854i)T \)
67 \( 1 + (-0.973 - 0.229i)T \)
71 \( 1 + (0.148 - 0.988i)T \)
73 \( 1 + (0.997 + 0.0660i)T \)
79 \( 1 + (-0.340 + 0.940i)T \)
83 \( 1 + (-0.431 - 0.901i)T \)
89 \( 1 + (0.909 - 0.416i)T \)
97 \( 1 + (-0.995 - 0.0990i)T \)
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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

−23.009038336837080713745880027087, −22.18586755248141776386362330870, −21.657306654427010187727947593379, −20.34289774683222162711341784394, −19.737604142415026180160985248795, −19.08657051966718381843258784471, −18.395063513277595885119968539190, −17.55240597787563015577979267788, −16.36659580796674472418847553583, −15.82402967320896697079461038290, −15.1814124394881940777329700895, −13.76266164646899289979471571263, −12.53724704089289310717565667148, −12.1324765099636751904533372428, −11.00536331370863879302361966098, −10.451383685434265673877911180305, −9.29426739745350042820573241348, −8.513955380509817535961023352885, −7.6852792761553722774975562147, −6.77643675518291316517944860943, −5.73253623207784413969220741041, −4.10811849010713023662270362423, −3.09499481145258366740260753216, −2.48298894353938930036076703686, −0.62840739943445533009830781013, 0.12444923388933099499883781014, 1.42741265695962544338124886298, 2.8042804403431070702368498461, 4.17127490079488513447139604548, 5.09342334140242183853440372901, 6.49655116895482650447426319785, 7.32516770455543586484465604084, 7.75545737479476300860373737444, 9.19629498461931705529519251744, 9.57595728453410789814299327673, 10.69437921611095766269054915167, 11.62310804582551406008610285656, 12.42865438540403148598917176053, 13.61856792102586318888275041549, 14.658086021059115523365325403872, 15.64642288453601401908604755101, 16.0816004400804304883223160525, 16.92018808015884907140661463293, 17.780713947105619409156156335454, 18.69991073627982624772882164926, 19.65797314277537203538400038235, 19.9681742819112018749361962033, 20.77238508831593123079219559286, 22.32650008664947081436395343325, 22.99940651367740310775935837451

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