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Properties

Label 1-569-569.5-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.138 + 0.990i$
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  + (0.154 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05456112897 + 0.06272130462i\)
\(L(\frac12)\) \(\approx\) \(0.05456112897 + 0.06272130462i\)
\(L(1)\) \(\approx\) \(0.5252367175 - 0.2430048246i\)
\(L(1)\) \(\approx\) \(0.5252367175 - 0.2430048246i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.154 - 0.988i)T \)
3 \( 1 + (-0.991 - 0.132i)T \)
5 \( 1 + (-0.0221 + 0.999i)T \)
7 \( 1 + (-0.367 - 0.930i)T \)
11 \( 1 + (0.487 + 0.873i)T \)
13 \( 1 + (0.325 - 0.945i)T \)
17 \( 1 + (-0.666 - 0.745i)T \)
19 \( 1 + (-0.952 + 0.304i)T \)
23 \( 1 + (0.240 - 0.970i)T \)
29 \( 1 + (-0.666 + 0.745i)T \)
31 \( 1 + (-0.598 + 0.801i)T \)
37 \( 1 + (-0.367 - 0.930i)T \)
41 \( 1 + (0.240 - 0.970i)T \)
43 \( 1 + (0.154 + 0.988i)T \)
47 \( 1 + (0.408 + 0.912i)T \)
53 \( 1 + (-0.283 + 0.958i)T \)
59 \( 1 + (0.408 + 0.912i)T \)
61 \( 1 + (0.0663 - 0.997i)T \)
67 \( 1 + (-0.666 + 0.745i)T \)
71 \( 1 + (-0.448 - 0.894i)T \)
73 \( 1 + (-0.952 + 0.304i)T \)
79 \( 1 + (-0.839 + 0.544i)T \)
83 \( 1 + (-0.730 + 0.683i)T \)
89 \( 1 + (-0.598 - 0.801i)T \)
97 \( 1 + (-0.999 - 0.0442i)T \)
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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

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

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