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Properties

Label 1-571-571.241-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.901 - 0.433i$
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  + (−0.245 + 0.969i)2-s + (−0.490 − 0.871i)3-s + (−0.879 − 0.475i)4-s + (0.894 + 0.446i)5-s + (0.965 − 0.261i)6-s + (−0.828 + 0.560i)7-s + (0.677 − 0.735i)8-s + (−0.518 + 0.854i)9-s + (−0.652 + 0.757i)10-s + (0.652 + 0.757i)11-s + (0.0165 + 0.999i)12-s + (−0.213 + 0.976i)13-s + (−0.340 − 0.940i)14-s + (−0.0495 − 0.998i)15-s + (0.546 + 0.837i)16-s + (0.934 + 0.355i)17-s + ⋯
L(s)  = 1  + (−0.245 + 0.969i)2-s + (−0.490 − 0.871i)3-s + (−0.879 − 0.475i)4-s + (0.894 + 0.446i)5-s + (0.965 − 0.261i)6-s + (−0.828 + 0.560i)7-s + (0.677 − 0.735i)8-s + (−0.518 + 0.854i)9-s + (−0.652 + 0.757i)10-s + (0.652 + 0.757i)11-s + (0.0165 + 0.999i)12-s + (−0.213 + 0.976i)13-s + (−0.340 − 0.940i)14-s + (−0.0495 − 0.998i)15-s + (0.546 + 0.837i)16-s + (0.934 + 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.901 - 0.433i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.901 - 0.433i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1569753807 + 0.6888381806i\)
\(L(\frac12)\) \(\approx\) \(-0.1569753807 + 0.6888381806i\)
\(L(1)\) \(\approx\) \(0.6229768894 + 0.3801701769i\)
\(L(1)\) \(\approx\) \(0.6229768894 + 0.3801701769i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.245 + 0.969i)T \)
3 \( 1 + (-0.490 - 0.871i)T \)
5 \( 1 + (0.894 + 0.446i)T \)
7 \( 1 + (-0.828 + 0.560i)T \)
11 \( 1 + (0.652 + 0.757i)T \)
13 \( 1 + (-0.213 + 0.976i)T \)
17 \( 1 + (0.934 + 0.355i)T \)
19 \( 1 + (-0.980 + 0.197i)T \)
23 \( 1 + (-0.909 + 0.416i)T \)
29 \( 1 + (-0.401 - 0.915i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (0.863 - 0.504i)T \)
41 \( 1 + (0.986 - 0.164i)T \)
43 \( 1 + (-0.973 + 0.229i)T \)
47 \( 1 + (-0.546 - 0.837i)T \)
53 \( 1 + (0.768 + 0.639i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.846 - 0.533i)T \)
67 \( 1 + (0.768 + 0.639i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.863 + 0.504i)T \)
79 \( 1 + (-0.180 - 0.983i)T \)
83 \( 1 + (0.965 - 0.261i)T \)
89 \( 1 + (-0.965 + 0.261i)T \)
97 \( 1 + (0.431 + 0.901i)T \)
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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

−22.30844929202996648086556980180, −21.729376982808268310157024270635, −20.90029046900068272671187055322, −20.2287208907499321643528160593, −19.483179502454770592780671112631, −18.32928293913810801014330309964, −17.47453266375536750372308724644, −16.67635419622920744481162306405, −16.38925213581009623365811860417, −14.79670751383280823278989713613, −13.91021294749375226154514784081, −12.989542247701823707493152696135, −12.29049058506553552761887059980, −11.21010472762240178136907418743, −10.31538484532218756116309651281, −9.84156734146719766158440796291, −9.09629962513246417481270391350, −8.100204824924836127031825267274, −6.388705957038884724080087843571, −5.58062355899387437798021319766, −4.52237297715424853554991441755, −3.584084551000009545665252357172, −2.70328987256107670835630594181, −1.06478243736283938625582901024, −0.236251143734314566192034901533, 1.41011886346885872745121562533, 2.33822054162366299881690708097, 4.05969497998921423934863448447, 5.41696826469408428586261856801, 6.19227592173713508409803571301, 6.63724030642816430491318023804, 7.52840305983320705141800649684, 8.71060098792294296957260516308, 9.65629576339551217873619016628, 10.27722307615070357283044253411, 11.75280474652703004773168831276, 12.63500915403947731323701718685, 13.41695231355982410715807684267, 14.30567951978900504030157538947, 14.929340584641502969811281731344, 16.249614922703498095247984526359, 16.91793442581407130913106431657, 17.577606197615200851645012473317, 18.387641520228208247661412953075, 19.06598820291735649817846713696, 19.659437636539586490171218479300, 21.50184979032356381915968112995, 22.02274237869178807610986696491, 23.04889709460212163802323015568, 23.339725498531849217621663299817

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