Properties

Degree 1
Conductor 569
Sign $-0.138 - 0.990i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.154 + 0.988i)2-s + (−0.991 + 0.132i)3-s + (−0.952 + 0.304i)4-s + (−0.0221 − 0.999i)5-s + (−0.283 − 0.958i)6-s + (−0.367 + 0.930i)7-s + (−0.448 − 0.894i)8-s + (0.964 − 0.262i)9-s + (0.984 − 0.176i)10-s + (0.487 − 0.873i)11-s + (0.903 − 0.428i)12-s + (0.325 + 0.945i)13-s + (−0.975 − 0.219i)14-s + (0.154 + 0.988i)15-s + (0.814 − 0.580i)16-s + (−0.666 + 0.745i)17-s + ⋯
L(s,χ)  = 1  + (0.154 + 0.988i)2-s + (−0.991 + 0.132i)3-s + (−0.952 + 0.304i)4-s + (−0.0221 − 0.999i)5-s + (−0.283 − 0.958i)6-s + (−0.367 + 0.930i)7-s + (−0.448 − 0.894i)8-s + (0.964 − 0.262i)9-s + (0.984 − 0.176i)10-s + (0.487 − 0.873i)11-s + (0.903 − 0.428i)12-s + (0.325 + 0.945i)13-s + (−0.975 − 0.219i)14-s + (0.154 + 0.988i)15-s + (0.814 − 0.580i)16-s + (−0.666 + 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.138 - 0.990i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.138 - 0.990i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.138 - 0.990i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (114, \cdot )$
Sato-Tate  :  $\mu(71)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ -0.138 - 0.990i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.05456112897 - 0.06272130462i$
$L(\frac12,\chi)$  $\approx$  $0.05456112897 - 0.06272130462i$
$L(\chi,1)$  $\approx$  0.5252367175 + 0.2430048246i
$L(1,\chi)$  $\approx$  0.5252367175 + 0.2430048246i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−23.02186138740674295776260207542, −22.65966430065891991266846542204, −22.14743892454067991754337708282, −20.99871105003409829993264249744, −20.15123443733409729299462933908, −19.34712920706585441671270758767, −18.4274546147462816556182666990, −17.779512243220727198881016856483, −17.168016194276468756309742624247, −15.96137349154288842915278418915, −14.83866965907758062066529555500, −14.0355748678374453794827228422, −12.91477306719440564660285753244, −12.47722487251391890882028140275, −11.25323216517055063060245737037, −10.66128882707556818554721956765, −10.23234016635197563312043274512, −9.12022747016459609133142031546, −7.516165997483598884633027185592, −6.76504065629397275523508448688, −5.784107027230375981689416338669, −4.54589251838281690685251895188, −3.825483375383184003275919898941, −2.61943723677298988760540203317, −1.380536710849999581382649750140, 0.04959359954009113918911267212, 1.65072681606669956967010441433, 3.77014476019653612821727148063, 4.47900379526268950844086257357, 5.59807627103127903041605148168, 6.04989170672051785217187508108, 6.926821000281543684162170260581, 8.37206978498610955359532397280, 8.98679093931956568851105160181, 9.746008380894798896406305879639, 11.30709140319287325333735847374, 11.96648930817350529125189685761, 13.022239905820862131773673022794, 13.421034957290193964530061544162, 14.958673890079388264838898746538, 15.60863328870892448426543832108, 16.42219330801198961018823103555, 16.93742701242487848972821245368, 17.63368856397584272255601105678, 18.795906697706095006764777960896, 19.28608638182348943454233911533, 20.987931282337037593620103240613, 21.761248062595076670793929390438, 22.087743131590737912618651769215, 23.296060807226166617338237931025

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