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Properties

Label 1-571-571.51-r0-0-0
Degree $1$
Conductor $571$
Sign $0.0129 + 0.999i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (0.546 − 0.837i)2-s + (−0.148 − 0.988i)3-s + (−0.401 − 0.915i)4-s + (−0.999 − 0.0330i)5-s + (−0.909 − 0.416i)6-s + (0.180 − 0.983i)7-s + (−0.986 − 0.164i)8-s + (−0.956 + 0.293i)9-s + (−0.574 + 0.818i)10-s + (−0.574 − 0.818i)11-s + (−0.846 + 0.533i)12-s + (−0.518 − 0.854i)13-s + (−0.724 − 0.689i)14-s + (0.115 + 0.993i)15-s + (−0.677 + 0.735i)16-s + (0.980 − 0.197i)17-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)2-s + (−0.148 − 0.988i)3-s + (−0.401 − 0.915i)4-s + (−0.999 − 0.0330i)5-s + (−0.909 − 0.416i)6-s + (0.180 − 0.983i)7-s + (−0.986 − 0.164i)8-s + (−0.956 + 0.293i)9-s + (−0.574 + 0.818i)10-s + (−0.574 − 0.818i)11-s + (−0.846 + 0.533i)12-s + (−0.518 − 0.854i)13-s + (−0.724 − 0.689i)14-s + (0.115 + 0.993i)15-s + (−0.677 + 0.735i)16-s + (0.980 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0129 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0129 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.0129 + 0.999i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.0129 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5953489325 - 0.6031256359i\)
\(L(\frac12)\) \(\approx\) \(-0.5953489325 - 0.6031256359i\)
\(L(1)\) \(\approx\) \(0.3626079463 - 0.8320851693i\)
\(L(1)\) \(\approx\) \(0.3626079463 - 0.8320851693i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.546 - 0.837i)T \)
3 \( 1 + (-0.148 - 0.988i)T \)
5 \( 1 + (-0.999 - 0.0330i)T \)
7 \( 1 + (0.180 - 0.983i)T \)
11 \( 1 + (-0.574 - 0.818i)T \)
13 \( 1 + (-0.518 - 0.854i)T \)
17 \( 1 + (0.980 - 0.197i)T \)
19 \( 1 + (0.894 - 0.446i)T \)
23 \( 1 + (-0.461 - 0.887i)T \)
29 \( 1 + (-0.0825 + 0.996i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (0.652 + 0.757i)T \)
41 \( 1 + (0.789 + 0.614i)T \)
43 \( 1 + (-0.0165 - 0.999i)T \)
47 \( 1 + (-0.677 + 0.735i)T \)
53 \( 1 + (0.0495 - 0.998i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (0.965 + 0.261i)T \)
67 \( 1 + (0.0495 - 0.998i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.652 + 0.757i)T \)
79 \( 1 + (-0.995 - 0.0990i)T \)
83 \( 1 + (-0.909 - 0.416i)T \)
89 \( 1 + (-0.909 - 0.416i)T \)
97 \( 1 + (0.863 + 0.504i)T \)
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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

−23.71490325783424845571410433197, −23.004132802704974206829775776576, −22.38351442234055762293626235814, −21.462960793023977503408928833471, −20.938665132368533425427320445231, −19.87221703532050729816823040687, −18.70777086966800555699271689082, −17.85955852269431903536392961458, −16.77875659552926521126434186688, −16.09815857544468851997671376979, −15.439129950091256105529577863712, −14.83513363542672147446974304066, −14.170528853772244598136190505588, −12.67273986465020167885053497996, −11.912188590672760044890645090406, −11.40317128419355986911040497825, −9.81704921713888508284650558209, −9.166902264484902106186600657120, −7.97859807846628873224686118849, −7.46630498560848223818385170065, −5.9795845341614799113160206425, −5.2563585542380360312255769339, −4.38718678024210619109946201932, −3.599478496123675028658217294867, −2.498359165288285436893623132, 0.41159020584231802233170838700, 1.22577324455956338920220631412, 2.86535564456079766402140974208, 3.41509359236902296273669129212, 4.79145997196021686953428706622, 5.5631518720177301289472952910, 6.87515329437043728487588339444, 7.73556484050699767057933263267, 8.494358378754753183058027732563, 10.01853666919263375363600416571, 10.94191088517993446692740092299, 11.48851693073236419710345518060, 12.516965255044419494602549830006, 12.9743963564913024798583607107, 14.08625425372315655027384838147, 14.53500473644993105008322506268, 15.88306480166662335209473123005, 16.76433506879504114740595632950, 18.031814554975739153353382398075, 18.561285870898928881378896588954, 19.51749792879062887925498357668, 20.0621622206203333738478201205, 20.64614406899126462454335937262, 21.98893892176116641022287507201, 22.79382203064352402816622413103

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