Properties

Degree 1
Conductor 569
Sign $0.710 + 0.703i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.387 + 0.921i)2-s + (0.438 − 0.898i)3-s + (−0.699 − 0.714i)4-s + (−0.666 + 0.745i)5-s + (0.658 + 0.752i)6-s + (−0.988 − 0.154i)7-s + (0.930 − 0.367i)8-s + (−0.616 − 0.787i)9-s + (−0.428 − 0.903i)10-s + (0.955 − 0.294i)11-s + (−0.948 + 0.315i)12-s + (0.0442 + 0.999i)13-s + (0.525 − 0.850i)14-s + (0.377 + 0.925i)15-s + (−0.0221 + 0.999i)16-s + (−0.506 − 0.862i)17-s + ⋯
L(s,χ)  = 1  + (−0.387 + 0.921i)2-s + (0.438 − 0.898i)3-s + (−0.699 − 0.714i)4-s + (−0.666 + 0.745i)5-s + (0.658 + 0.752i)6-s + (−0.988 − 0.154i)7-s + (0.930 − 0.367i)8-s + (−0.616 − 0.787i)9-s + (−0.428 − 0.903i)10-s + (0.955 − 0.294i)11-s + (−0.948 + 0.315i)12-s + (0.0442 + 0.999i)13-s + (0.525 − 0.850i)14-s + (0.377 + 0.925i)15-s + (−0.0221 + 0.999i)16-s + (−0.506 − 0.862i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.710 + 0.703i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.710 + 0.703i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.710 + 0.703i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (29, \cdot )$
Sato-Tate  :  $\mu(568)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (1:\ ),\ 0.710 + 0.703i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8223572548 + 0.3383514429i$
$L(\frac12,\chi)$  $\approx$  $0.8223572548 + 0.3383514429i$
$L(\chi,1)$  $\approx$  0.6887492664 + 0.1306201796i
$L(1,\chi)$  $\approx$  0.6887492664 + 0.1306201796i

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

−22.59751180820291584889959533535, −22.020449933040401164295817615496, −21.18514521016286505987177801558, −20.176748096082794141384314651, −19.72635928833609981177949715617, −19.368879316191943615023628618865, −17.98012475677503769910989945257, −16.902686345561403154745548850273, −16.40951099230861480887632879173, −15.429943566609427338424412075394, −14.57424832494533196387863058431, −13.25338497417020526691070878731, −12.64246861998459192417204865126, −11.81882674750274706289181600407, −10.66876162030071212050707360572, −10.08387312475106533216125466812, −8.95870641047407039395155577845, −8.687231411073003075731787475258, −7.60847886221758698002048198761, −6.01902498011407796072885306051, −4.61292624957655477951588706882, −3.96686150362264210643288215416, −3.21274787268320531631561099108, −1.99650773945515032305948149974, −0.4351671326208460558655815583, 0.57385185021322918292074080112, 2.03817161527280965979263909039, 3.42432121926597888498204521789, 4.289331098737028320629743184588, 6.10122398717638037514182186035, 6.67950463745208266786836143035, 7.15094740152775475826367452273, 8.2473854830451123732048808153, 9.05881640492692711419597753656, 9.879797252413444272642021271451, 11.21343438077114636119790344033, 12.0782491266298538152035191521, 13.262805343888066945977008911758, 14.02416926267951013379025015068, 14.61059771833614087272061964776, 15.61662890589388924201404814507, 16.38629181993467821783553653508, 17.314034389452670180202779528410, 18.27198851425485516333126063669, 19.01167791335219581566995925583, 19.44565935978512020649731002641, 20.11338216043621025924524222278, 21.906946049146468907479591071455, 22.576007625799306976033866889759, 23.50855235242290748340686902177

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