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Properties

Label 1-571-571.374-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.744 + 0.668i$
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  + (0.137 + 0.990i)2-s + (0.583 + 0.812i)3-s + (−0.962 + 0.272i)4-s + (−0.480 − 0.876i)5-s + (−0.724 + 0.689i)6-s + (0.601 − 0.799i)7-s + (−0.401 − 0.915i)8-s + (−0.319 + 0.947i)9-s + (0.802 − 0.596i)10-s + (−0.917 + 0.396i)11-s + (−0.782 − 0.622i)12-s + (0.775 + 0.631i)13-s + (0.874 + 0.485i)14-s + (0.431 − 0.901i)15-s + (0.851 − 0.523i)16-s + (−0.381 + 0.924i)17-s + ⋯
L(s)  = 1  + (0.137 + 0.990i)2-s + (0.583 + 0.812i)3-s + (−0.962 + 0.272i)4-s + (−0.480 − 0.876i)5-s + (−0.724 + 0.689i)6-s + (0.601 − 0.799i)7-s + (−0.401 − 0.915i)8-s + (−0.319 + 0.947i)9-s + (0.802 − 0.596i)10-s + (−0.917 + 0.396i)11-s + (−0.782 − 0.622i)12-s + (0.775 + 0.631i)13-s + (0.874 + 0.485i)14-s + (0.431 − 0.901i)15-s + (0.851 − 0.523i)16-s + (−0.381 + 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.744 + 0.668i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.744 + 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5059214034 + 1.320718133i\)
\(L(\frac12)\) \(\approx\) \(0.5059214034 + 1.320718133i\)
\(L(1)\) \(\approx\) \(0.8699840860 + 0.7684208603i\)
\(L(1)\) \(\approx\) \(0.8699840860 + 0.7684208603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.137 + 0.990i)T \)
3 \( 1 + (0.583 + 0.812i)T \)
5 \( 1 + (-0.480 - 0.876i)T \)
7 \( 1 + (0.601 - 0.799i)T \)
11 \( 1 + (-0.917 + 0.396i)T \)
13 \( 1 + (0.775 + 0.631i)T \)
17 \( 1 + (-0.381 + 0.924i)T \)
19 \( 1 + (0.952 - 0.303i)T \)
23 \( 1 + (0.746 + 0.665i)T \)
29 \( 1 + (0.451 + 0.892i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (0.840 + 0.542i)T \)
41 \( 1 + (0.904 + 0.426i)T \)
43 \( 1 + (-0.999 + 0.0110i)T \)
47 \( 1 + (0.0275 + 0.999i)T \)
53 \( 1 + (0.471 + 0.882i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (-0.644 + 0.764i)T \)
67 \( 1 + (0.528 - 0.849i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.889 + 0.456i)T \)
79 \( 1 + (-0.441 - 0.897i)T \)
83 \( 1 + (0.959 + 0.282i)T \)
89 \( 1 + (-0.234 - 0.972i)T \)
97 \( 1 + (0.938 + 0.345i)T \)
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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

−22.98478674240768642534214400352, −22.06121126712284939868352989259, −21.032810563765586279176162650678, −20.473808401123099387592283119503, −19.55209524187357509135142394314, −18.64761127658883879994872156818, −18.26540529110084529816429580514, −17.83148829409006385232507677294, −15.92131564324870516339544819035, −15.02196613387591910954126033254, −14.29616679920718293154622485806, −13.48423040980318116500637674212, −12.70171673966263127313025899876, −11.66670562809548314280171195794, −11.21563291697963836339626856129, −10.15770010944821783316943606596, −8.94054366984959077900485092166, −8.22761799762520497630803534918, −7.42943397635593499755888056946, −6.06063923182754997232170064251, −5.10943174990763403255500717718, −3.619322395671140428152517946365, −2.83172774045046792176359711387, −2.20720635752100829734828014705, −0.7279664670832002416565303613, 1.35182550415572144289807993526, 3.28081333990902112956804991904, 4.25565908863516521764508868765, 4.75419136645226857188428290212, 5.6781938357791534902059460980, 7.27225890730275080540206705207, 7.89265143430201619016457685368, 8.70363712230759315946381169625, 9.414892042629777943176002275848, 10.53381195319425904624716431569, 11.512920287647796098475891733822, 13.057855187845221610098709711009, 13.41716525788472798589022846691, 14.48462524392123920869987804421, 15.23952650708123379458528575903, 16.01088705612033166772457186447, 16.56246555633565743777491415122, 17.40895476589464050424033118638, 18.386930181481189945419282475682, 19.58859768871348228050853680789, 20.365099459789835593115439582839, 21.119668491812219332800550120015, 21.80329119445496096118782199806, 23.0671058039387079922341615500, 23.671496428886538322784935880736

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