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Properties

Label 1-569-569.29-r1-0-0
Degree $1$
Conductor $569$
Sign $0.710 + 0.703i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (−0.387 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.710 + 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8223572548 + 0.3383514429i\)
\(L(\frac12)\) \(\approx\) \(0.8223572548 + 0.3383514429i\)
\(L(1)\) \(\approx\) \(0.6887492664 + 0.1306201796i\)
\(L(1)\) \(\approx\) \(0.6887492664 + 0.1306201796i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.387 + 0.921i)T \)
3 \( 1 + (0.438 - 0.898i)T \)
5 \( 1 + (-0.666 + 0.745i)T \)
7 \( 1 + (-0.988 - 0.154i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (0.0442 + 0.999i)T \)
17 \( 1 + (-0.506 - 0.862i)T \)
19 \( 1 + (-0.999 + 0.0110i)T \)
23 \( 1 + (-0.820 + 0.571i)T \)
29 \( 1 + (0.967 - 0.251i)T \)
31 \( 1 + (-0.999 + 0.0331i)T \)
37 \( 1 + (0.589 + 0.807i)T \)
41 \( 1 + (-0.984 - 0.176i)T \)
43 \( 1 + (0.921 - 0.387i)T \)
47 \( 1 + (-0.867 + 0.496i)T \)
53 \( 1 + (0.752 - 0.658i)T \)
59 \( 1 + (-0.496 - 0.867i)T \)
61 \( 1 + (0.580 - 0.814i)T \)
67 \( 1 + (-0.506 + 0.862i)T \)
71 \( 1 + (-0.930 - 0.367i)T \)
73 \( 1 + (0.0110 + 0.999i)T \)
79 \( 1 + (0.132 + 0.991i)T \)
83 \( 1 + (0.888 - 0.457i)T \)
89 \( 1 + (-0.0331 + 0.999i)T \)
97 \( 1 + (0.780 + 0.624i)T \)
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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

−22.59751180820291584889959533535, −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