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Properties

Label 1-571-571.95-r1-0-0
Degree $1$
Conductor $571$
Sign $0.621 - 0.783i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (−0.945 + 0.324i)2-s + (−0.965 − 0.261i)3-s + (0.789 − 0.614i)4-s + (0.115 − 0.993i)5-s + (0.997 − 0.0660i)6-s + (0.148 + 0.988i)7-s + (−0.546 + 0.837i)8-s + (0.863 + 0.504i)9-s + (0.213 + 0.976i)10-s + (−0.213 + 0.976i)11-s + (−0.922 + 0.386i)12-s + (0.431 − 0.901i)13-s + (−0.461 − 0.887i)14-s + (−0.371 + 0.928i)15-s + (0.245 − 0.969i)16-s + (0.768 − 0.639i)17-s + ⋯
L(s)  = 1  + (−0.945 + 0.324i)2-s + (−0.965 − 0.261i)3-s + (0.789 − 0.614i)4-s + (0.115 − 0.993i)5-s + (0.997 − 0.0660i)6-s + (0.148 + 0.988i)7-s + (−0.546 + 0.837i)8-s + (0.863 + 0.504i)9-s + (0.213 + 0.976i)10-s + (−0.213 + 0.976i)11-s + (−0.922 + 0.386i)12-s + (0.431 − 0.901i)13-s + (−0.461 − 0.887i)14-s + (−0.371 + 0.928i)15-s + (0.245 − 0.969i)16-s + (0.768 − 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.621 - 0.783i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.621 - 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7371510513 - 0.3560602063i\)
\(L(\frac12)\) \(\approx\) \(0.7371510513 - 0.3560602063i\)
\(L(1)\) \(\approx\) \(0.5735426972 - 0.04646661808i\)
\(L(1)\) \(\approx\) \(0.5735426972 - 0.04646661808i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.945 + 0.324i)T \)
3 \( 1 + (-0.965 - 0.261i)T \)
5 \( 1 + (0.115 - 0.993i)T \)
7 \( 1 + (0.148 + 0.988i)T \)
11 \( 1 + (-0.213 + 0.976i)T \)
13 \( 1 + (0.431 - 0.901i)T \)
17 \( 1 + (0.768 - 0.639i)T \)
19 \( 1 + (-0.0495 - 0.998i)T \)
23 \( 1 + (-0.627 + 0.778i)T \)
29 \( 1 + (-0.879 + 0.475i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (0.991 - 0.131i)T \)
41 \( 1 + (0.677 - 0.735i)T \)
43 \( 1 + (0.746 + 0.665i)T \)
47 \( 1 + (-0.245 + 0.969i)T \)
53 \( 1 + (0.574 + 0.818i)T \)
59 \( 1 + (-0.986 - 0.164i)T \)
61 \( 1 + (0.601 + 0.799i)T \)
67 \( 1 + (0.574 + 0.818i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.991 + 0.131i)T \)
79 \( 1 + (0.340 - 0.940i)T \)
83 \( 1 + (0.997 - 0.0660i)T \)
89 \( 1 + (-0.997 + 0.0660i)T \)
97 \( 1 + (-0.277 + 0.960i)T \)
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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

−23.20322095091214683334427507391, −22.18551104373170943165490693836, −21.343130039618152083600630720201, −20.8949013224352362869655170292, −19.59460371526292947650551470360, −18.67699420104269224096446426047, −18.334134491390320412125936909914, −17.25704223752740660966383305777, −16.588370122968990585311947897929, −16.09046400948531968400206399673, −14.812384635941519481776127136257, −13.86876189609020987465687939286, −12.65301970109716760558614420604, −11.57073227098050130426136583137, −11.01499385155692326283244164192, −10.33246035754133834390043341212, −9.753566028925518822643817516140, −8.31578831746849153644548936727, −7.4184968550393402727543679722, −6.47816945931231964642856049988, −5.889106857251333879591854181624, −4.0641307644178346615690347027, −3.43707083284631424794946935101, −1.85769714320066244459875419190, −0.76048423533521757278383002464, 0.47434212378443393259823138737, 1.453805690792405743213162303092, 2.5096914413011935076670318407, 4.60130803777976513753872995056, 5.56150954436818884176740040385, 5.9322756494767638741449334712, 7.39887461845220819400016919172, 7.94462589367398077592472654627, 9.21963174052845746226637179686, 9.72152582145599013982911330811, 10.899147080446873921602652600768, 11.76287711990124719857112670668, 12.447468386321013568804059870838, 13.30062299846799617559164905505, 14.8928974923107818058122202888, 15.73857540430545880534491906867, 16.20429098058874793853096400230, 17.301731893395435073695028046206, 17.797318439580200732985106452855, 18.41523952879198981805445575923, 19.41008168323116409873418313929, 20.4034973053726198629540026664, 21.065316149672038315534532726493, 22.15565300196714246553482931409, 23.16475479610204123665276542911

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