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Properties

Label 1-571-571.187-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.601 - 0.799i$
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  + (0.677 + 0.735i)2-s + (0.340 + 0.940i)3-s + (−0.0825 + 0.996i)4-s + (0.431 + 0.901i)5-s + (−0.461 + 0.887i)6-s + (0.995 − 0.0990i)7-s + (−0.789 + 0.614i)8-s + (−0.768 + 0.639i)9-s + (−0.371 + 0.928i)10-s + (0.371 + 0.928i)11-s + (−0.965 + 0.261i)12-s + (−0.956 − 0.293i)13-s + (0.746 + 0.665i)14-s + (−0.701 + 0.712i)15-s + (−0.986 − 0.164i)16-s + (−0.894 + 0.446i)17-s + ⋯
L(s)  = 1  + (0.677 + 0.735i)2-s + (0.340 + 0.940i)3-s + (−0.0825 + 0.996i)4-s + (0.431 + 0.901i)5-s + (−0.461 + 0.887i)6-s + (0.995 − 0.0990i)7-s + (−0.789 + 0.614i)8-s + (−0.768 + 0.639i)9-s + (−0.371 + 0.928i)10-s + (0.371 + 0.928i)11-s + (−0.965 + 0.261i)12-s + (−0.956 − 0.293i)13-s + (0.746 + 0.665i)14-s + (−0.701 + 0.712i)15-s + (−0.986 − 0.164i)16-s + (−0.894 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.601 - 0.799i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.601 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.485765704 + 2.976809219i\)
\(L(\frac12)\) \(\approx\) \(-1.485765704 + 2.976809219i\)
\(L(1)\) \(\approx\) \(0.7958616337 + 1.612520290i\)
\(L(1)\) \(\approx\) \(0.7958616337 + 1.612520290i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.677 + 0.735i)T \)
3 \( 1 + (0.340 + 0.940i)T \)
5 \( 1 + (0.431 + 0.901i)T \)
7 \( 1 + (0.995 - 0.0990i)T \)
11 \( 1 + (0.371 + 0.928i)T \)
13 \( 1 + (-0.956 - 0.293i)T \)
17 \( 1 + (-0.894 + 0.446i)T \)
19 \( 1 + (0.999 - 0.0330i)T \)
23 \( 1 + (0.828 - 0.560i)T \)
29 \( 1 + (0.945 - 0.324i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.574 - 0.818i)T \)
41 \( 1 + (0.879 + 0.475i)T \)
43 \( 1 + (-0.846 + 0.533i)T \)
47 \( 1 + (0.986 + 0.164i)T \)
53 \( 1 + (-0.115 + 0.993i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (-0.909 + 0.416i)T \)
67 \( 1 + (-0.115 + 0.993i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.574 + 0.818i)T \)
79 \( 1 + (0.973 + 0.229i)T \)
83 \( 1 + (-0.461 + 0.887i)T \)
89 \( 1 + (0.461 - 0.887i)T \)
97 \( 1 + (0.652 - 0.757i)T \)
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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

−22.342858786357039862117640622998, −21.56797587270381727189872440395, −20.80264409419443209515458423199, −20.019964616038889297526800546711, −19.47335417333235723444422789804, −18.41022018373146139155469817137, −17.72093069466433327151986319887, −16.80350010397995394350222746653, −15.486358252148193338155758528693, −14.41322949592870690464945085511, −13.82249167399021649019707263519, −13.26299893476879231026421407233, −12.094806592054902528732874134827, −11.79545453526928956317688994788, −10.7337289072875231529517309727, −9.225045225565455134184954682982, −8.88939685260825878915827489274, −7.60442566908645257436143276050, −6.48876107485865333475426916473, −5.37063615453709747125988083013, −4.80454894332085343023942620728, −3.39343436991117894753838203747, −2.268397205028869331918538066416, −1.45358610922555039669445088564, −0.57848230801533445553479533497, 2.14378730605308120548053380883, 2.960260447844651127709289656937, 4.189350284189576327768812433286, 4.84611081638991301943211282895, 5.74990441488902424125166645231, 7.01656060039979308769616348521, 7.66042762890870096365670110628, 8.80886233741693199140969806402, 9.707791038509655298755696440356, 10.7571053408900475830180389297, 11.531397599072014142369322667308, 12.674443975558265298802954475, 13.91496242387439938254998513030, 14.42913883470479566159485494018, 15.04053444613322470012697108959, 15.61244116372248520421288452405, 16.92254244580015135975448071037, 17.48415755389213399224241030184, 18.179472441706373091170097912879, 19.73738390753142217793437484528, 20.4963639745986264991927434399, 21.37884735489731479190043513382, 22.00747538395223480402117655810, 22.58750376771464688286443052557, 23.384617190023345170684690266319

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