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Properties

Label 1-569-569.93-r0-0-0
Degree $1$
Conductor $569$
Sign $0.569 + 0.821i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (0.759 + 0.650i)2-s + (−0.0663 − 0.997i)3-s + (0.154 + 0.988i)4-s + (0.699 − 0.714i)5-s + (0.598 − 0.801i)6-s + (0.562 + 0.826i)7-s + (−0.525 + 0.850i)8-s + (−0.991 + 0.132i)9-s + (0.996 − 0.0883i)10-s + (−0.862 + 0.506i)11-s + (0.975 − 0.219i)12-s + (0.814 + 0.580i)13-s + (−0.110 + 0.993i)14-s + (−0.759 − 0.650i)15-s + (−0.952 + 0.304i)16-s + (0.408 + 0.912i)17-s + ⋯
L(s)  = 1  + (0.759 + 0.650i)2-s + (−0.0663 − 0.997i)3-s + (0.154 + 0.988i)4-s + (0.699 − 0.714i)5-s + (0.598 − 0.801i)6-s + (0.562 + 0.826i)7-s + (−0.525 + 0.850i)8-s + (−0.991 + 0.132i)9-s + (0.996 − 0.0883i)10-s + (−0.862 + 0.506i)11-s + (0.975 − 0.219i)12-s + (0.814 + 0.580i)13-s + (−0.110 + 0.993i)14-s + (−0.759 − 0.650i)15-s + (−0.952 + 0.304i)16-s + (0.408 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.019299655 + 1.057131855i\)
\(L(\frac12)\) \(\approx\) \(2.019299655 + 1.057131855i\)
\(L(1)\) \(\approx\) \(1.645182909 + 0.4322360475i\)
\(L(1)\) \(\approx\) \(1.645182909 + 0.4322360475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.759 + 0.650i)T \)
3 \( 1 + (-0.0663 - 0.997i)T \)
5 \( 1 + (0.699 - 0.714i)T \)
7 \( 1 + (0.562 + 0.826i)T \)
11 \( 1 + (-0.862 + 0.506i)T \)
13 \( 1 + (0.814 + 0.580i)T \)
17 \( 1 + (0.408 + 0.912i)T \)
19 \( 1 + (-0.154 + 0.988i)T \)
23 \( 1 + (0.787 + 0.616i)T \)
29 \( 1 + (-0.408 + 0.912i)T \)
31 \( 1 + (0.448 - 0.894i)T \)
37 \( 1 + (-0.562 - 0.826i)T \)
41 \( 1 + (-0.787 - 0.616i)T \)
43 \( 1 + (0.759 - 0.650i)T \)
47 \( 1 + (0.839 - 0.544i)T \)
53 \( 1 + (0.598 - 0.801i)T \)
59 \( 1 + (0.839 - 0.544i)T \)
61 \( 1 + (-0.730 - 0.683i)T \)
67 \( 1 + (0.408 - 0.912i)T \)
71 \( 1 + (-0.525 - 0.850i)T \)
73 \( 1 + (-0.154 + 0.988i)T \)
79 \( 1 + (-0.283 + 0.958i)T \)
83 \( 1 + (0.367 - 0.930i)T \)
89 \( 1 + (0.448 + 0.894i)T \)
97 \( 1 + (0.0221 + 0.999i)T \)
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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

−22.95851111363017582242946943580, −22.256270646286338666437191307425, −21.28976170065174222510662771220, −20.88998267452582926073303658585, −20.284672115156574516213447875520, −19.078123177252560667646753208297, −18.16676077161796722873956218434, −17.32914573701447402413441134273, −16.163347366129175039199451117504, −15.3311450211588190718737533655, −14.5897966460113104557160359796, −13.65937879095330555290049034700, −13.344834272709266441196992760651, −11.709798960302886233738979520489, −10.84045847170759738455378965664, −10.583505575477424438094066412636, −9.72000753326652176113659056841, −8.59488301687537957213858891815, −7.138667196197818934883413019616, −5.99434736228206637288298087423, −5.19979604173135667891901906101, −4.39379946479121962395667027980, −3.1573847002139878161170265832, −2.68568442986373985962776929291, −0.96903430595688242181800332103, 1.64868861338199378740001217607, 2.29452264423276436025011253175, 3.75813915311345015229242230773, 5.23575138696982786461835290341, 5.58187963429712303848652463468, 6.49605375232281318379442039908, 7.63970875330936938912719577699, 8.39906339486741777597198658755, 9.07499137678792679464969115717, 10.74536340588097807939839702139, 11.89948249314243730373861429819, 12.53369709589673530720499579895, 13.15827728139966761227750843992, 13.968469119931203825668384898392, 14.77963965219134430896275997398, 15.739739620931348333495977768600, 16.79514030618246489137542008664, 17.40802681200826370996823149299, 18.23778620070065706778257432814, 18.9373236432333597541404001919, 20.46428145072098841794751774148, 20.9883469124972227823114999541, 21.693145296989256796525908235301, 22.822127403236912575362152210084, 23.64432141941188881404479817493

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