Properties

Label 1-569-569.49-r0-0-0
Degree $1$
Conductor $569$
Sign $0.793 + 0.608i$
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  + (−0.525 − 0.850i)2-s + (0.197 − 0.980i)3-s + (−0.448 + 0.894i)4-s + (−0.730 + 0.683i)5-s + (−0.937 + 0.346i)6-s + (−0.975 − 0.219i)7-s + (0.996 − 0.0883i)8-s + (−0.921 − 0.387i)9-s + (0.964 + 0.262i)10-s + (0.0221 − 0.999i)11-s + (0.787 + 0.616i)12-s + (−0.283 − 0.958i)13-s + (0.325 + 0.945i)14-s + (0.525 + 0.850i)15-s + (−0.598 − 0.801i)16-s + (−0.952 + 0.304i)17-s + ⋯
L(s)  = 1  + (−0.525 − 0.850i)2-s + (0.197 − 0.980i)3-s + (−0.448 + 0.894i)4-s + (−0.730 + 0.683i)5-s + (−0.937 + 0.346i)6-s + (−0.975 − 0.219i)7-s + (0.996 − 0.0883i)8-s + (−0.921 − 0.387i)9-s + (0.964 + 0.262i)10-s + (0.0221 − 0.999i)11-s + (0.787 + 0.616i)12-s + (−0.283 − 0.958i)13-s + (0.325 + 0.945i)14-s + (0.525 + 0.850i)15-s + (−0.598 − 0.801i)16-s + (−0.952 + 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1731067694 + 0.05868659629i\)
\(L(\frac12)\) \(\approx\) \(0.1731067694 + 0.05868659629i\)
\(L(1)\) \(\approx\) \(0.4306660114 - 0.2838887111i\)
\(L(1)\) \(\approx\) \(0.4306660114 - 0.2838887111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.525 - 0.850i)T \)
3 \( 1 + (0.197 - 0.980i)T \)
5 \( 1 + (-0.730 + 0.683i)T \)
7 \( 1 + (-0.975 - 0.219i)T \)
11 \( 1 + (0.0221 - 0.999i)T \)
13 \( 1 + (-0.283 - 0.958i)T \)
17 \( 1 + (-0.952 + 0.304i)T \)
19 \( 1 + (0.448 + 0.894i)T \)
23 \( 1 + (-0.408 - 0.912i)T \)
29 \( 1 + (0.952 + 0.304i)T \)
31 \( 1 + (-0.984 - 0.176i)T \)
37 \( 1 + (0.975 + 0.219i)T \)
41 \( 1 + (0.408 + 0.912i)T \)
43 \( 1 + (-0.525 + 0.850i)T \)
47 \( 1 + (-0.154 + 0.988i)T \)
53 \( 1 + (-0.937 + 0.346i)T \)
59 \( 1 + (-0.154 + 0.988i)T \)
61 \( 1 + (0.633 + 0.773i)T \)
67 \( 1 + (-0.952 - 0.304i)T \)
71 \( 1 + (0.996 + 0.0883i)T \)
73 \( 1 + (0.448 + 0.894i)T \)
79 \( 1 + (0.759 + 0.650i)T \)
83 \( 1 + (-0.903 - 0.428i)T \)
89 \( 1 + (-0.984 + 0.176i)T \)
97 \( 1 + (-0.0663 + 0.997i)T \)
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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

−23.303001864617379174887016601505, −22.415974539342974212735662154218, −21.73403297511087857441562844196, −20.307179056426810740812748630, −19.818716442865359347775631235476, −19.21841949473093613387261192138, −17.92416520502961892188000717672, −17.04188127335755309380939175839, −16.26017835255071692836975006184, −15.65431904955713343882408623933, −15.23836510159808540077772317473, −14.08531863577018163604652414065, −13.16485978549407142401031689743, −11.941227497096400012218019778348, −10.98541542406023104856748770174, −9.682728594437905608017599478192, −9.37267597959152754355187334377, −8.602498420883256152164841639335, −7.438086225616815573024830106998, −6.62665132880111406310851228979, −5.29369102788501220862496982782, −4.59094194417145592839596376622, −3.73848229691945342116015856092, −2.138775878339233175529190798870, −0.129215143816941581353281521387, 1.050106771974739236484709983310, 2.69207582831069692190691551258, 3.085097135739617619073262480772, 4.12787954306191801681353083052, 6.00343359327106536693978770487, 6.85567020323793031816720379279, 7.88987752381930377034188773015, 8.37408766833308207533834128876, 9.581795828267306095598547796097, 10.63324337209393243524141947782, 11.28785946267788274967802677371, 12.32023806201190784724390567492, 12.872829578648621551083895111216, 13.75735811495649016339908412217, 14.68399127227441407598470505208, 16.02836611261658021388966886806, 16.74188830560614774949771188107, 18.010424728756346407288392538078, 18.40202650683186141632509149971, 19.32030077610210333336246013647, 19.76893790276753039570796230867, 20.37321631911428554936047390377, 21.83167298994814262826872437069, 22.52416452419983214805318462039, 23.09766463808722991305965036960

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