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Properties

Label 1-569-569.14-r0-0-0
Degree $1$
Conductor $569$
Sign $0.352 + 0.935i$
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  + (−0.0221 + 0.999i)2-s + (−0.240 + 0.970i)3-s + (−0.999 − 0.0442i)4-s + (−0.975 + 0.219i)5-s + (−0.964 − 0.262i)6-s + (0.814 + 0.580i)7-s + (0.0663 − 0.997i)8-s + (−0.883 − 0.467i)9-s + (−0.197 − 0.980i)10-s + (0.367 − 0.930i)11-s + (0.283 − 0.958i)12-s + (0.984 − 0.176i)13-s + (−0.598 + 0.801i)14-s + (0.0221 − 0.999i)15-s + (0.996 + 0.0883i)16-s + (−0.525 − 0.850i)17-s + ⋯
L(s)  = 1  + (−0.0221 + 0.999i)2-s + (−0.240 + 0.970i)3-s + (−0.999 − 0.0442i)4-s + (−0.975 + 0.219i)5-s + (−0.964 − 0.262i)6-s + (0.814 + 0.580i)7-s + (0.0663 − 0.997i)8-s + (−0.883 − 0.467i)9-s + (−0.197 − 0.980i)10-s + (0.367 − 0.930i)11-s + (0.283 − 0.958i)12-s + (0.984 − 0.176i)13-s + (−0.598 + 0.801i)14-s + (0.0221 − 0.999i)15-s + (0.996 + 0.0883i)16-s + (−0.525 − 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.352 + 0.935i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.352 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7816036108 + 0.5405189254i\)
\(L(\frac12)\) \(\approx\) \(0.7816036108 + 0.5405189254i\)
\(L(1)\) \(\approx\) \(0.6727849212 + 0.5006167660i\)
\(L(1)\) \(\approx\) \(0.6727849212 + 0.5006167660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.0221 + 0.999i)T \)
3 \( 1 + (-0.240 + 0.970i)T \)
5 \( 1 + (-0.975 + 0.219i)T \)
7 \( 1 + (0.814 + 0.580i)T \)
11 \( 1 + (0.367 - 0.930i)T \)
13 \( 1 + (0.984 - 0.176i)T \)
17 \( 1 + (-0.525 - 0.850i)T \)
19 \( 1 + (0.999 - 0.0442i)T \)
23 \( 1 + (-0.759 - 0.650i)T \)
29 \( 1 + (0.525 - 0.850i)T \)
31 \( 1 + (0.991 - 0.132i)T \)
37 \( 1 + (-0.814 - 0.580i)T \)
41 \( 1 + (0.759 + 0.650i)T \)
43 \( 1 + (-0.0221 - 0.999i)T \)
47 \( 1 + (-0.487 - 0.873i)T \)
53 \( 1 + (-0.964 - 0.262i)T \)
59 \( 1 + (-0.487 - 0.873i)T \)
61 \( 1 + (-0.787 + 0.616i)T \)
67 \( 1 + (-0.525 + 0.850i)T \)
71 \( 1 + (0.0663 + 0.997i)T \)
73 \( 1 + (0.999 - 0.0442i)T \)
79 \( 1 + (0.862 - 0.506i)T \)
83 \( 1 + (-0.325 - 0.945i)T \)
89 \( 1 + (0.991 + 0.132i)T \)
97 \( 1 + (-0.903 + 0.428i)T \)
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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

−22.99818236437625710809969323837, −22.55293786242350888097036523179, −21.228678395374583500779165843892, −20.30569109532956872422068083682, −19.8367788281976848837539415992, −19.127762761377721468905901833686, −18.00642084377039688701925700619, −17.71384535887952093143787361297, −16.6801698077791594641046332723, −15.41259261625587507564886537573, −14.22312272972339636611416729857, −13.65327247371553090737413661769, −12.56622525567459705169897339806, −12.000010094739327532425302965203, −11.23813940534408971179585720010, −10.59955266375286632384718383599, −9.17249847242659782380069930608, −8.15635116684004668518666234708, −7.68156639264716637889801955062, −6.45555889662234161967721858132, −5.028065130255936068552779509360, −4.22921948684760806758510312251, −3.22077883379241052828772121289, −1.68637210608940526581365519320, −1.15121616627559315683697940073, 0.656458474282766753225383486957, 3.04516718597042360749194459582, 3.99686158345566495535042400132, 4.78592324876977648676606798391, 5.72231779679025548708665791207, 6.5836698912341188195765396171, 7.96535327522779702447696779012, 8.500963928273332671750775563152, 9.28901160174176412809899077332, 10.53623947709955363879753450965, 11.449941228462348701460743442932, 12.04338907046770153724453772007, 13.72393221026827467029954859805, 14.307853820486058004201820557652, 15.274598503149642371738558461129, 15.89934389996509481878908126700, 16.260840361342538334135785925882, 17.48373250328356502935002747552, 18.21157032265308544275765791140, 19.017977271781693662113169156747, 20.1948650134130569866846086850, 21.10011664283248805386707290616, 22.03548333255810861605890926215, 22.66069756394540166413167118548, 23.34213354411291886232537904636

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