Properties

Label 1-569-569.245-r0-0-0
Degree $1$
Conductor $569$
Sign $0.691 - 0.722i$
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  + (−0.921 + 0.387i)2-s + (−0.325 + 0.945i)3-s + (0.699 − 0.714i)4-s + (−0.666 − 0.745i)5-s + (−0.0663 − 0.997i)6-s + (0.154 + 0.988i)7-s + (−0.367 + 0.930i)8-s + (−0.787 − 0.616i)9-s + (0.903 + 0.428i)10-s + (0.883 − 0.467i)11-s + (0.448 + 0.894i)12-s + (−0.999 − 0.0442i)13-s + (−0.525 − 0.850i)14-s + (0.921 − 0.387i)15-s + (−0.0221 − 0.999i)16-s + (0.862 + 0.506i)17-s + ⋯
L(s)  = 1  + (−0.921 + 0.387i)2-s + (−0.325 + 0.945i)3-s + (0.699 − 0.714i)4-s + (−0.666 − 0.745i)5-s + (−0.0663 − 0.997i)6-s + (0.154 + 0.988i)7-s + (−0.367 + 0.930i)8-s + (−0.787 − 0.616i)9-s + (0.903 + 0.428i)10-s + (0.883 − 0.467i)11-s + (0.448 + 0.894i)12-s + (−0.999 − 0.0442i)13-s + (−0.525 − 0.850i)14-s + (0.921 − 0.387i)15-s + (−0.0221 − 0.999i)16-s + (0.862 + 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.691 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3422355784 - 0.1460794084i\)
\(L(\frac12)\) \(\approx\) \(0.3422355784 - 0.1460794084i\)
\(L(1)\) \(\approx\) \(0.4864892781 + 0.1212340219i\)
\(L(1)\) \(\approx\) \(0.4864892781 + 0.1212340219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.921 + 0.387i)T \)
3 \( 1 + (-0.325 + 0.945i)T \)
5 \( 1 + (-0.666 - 0.745i)T \)
7 \( 1 + (0.154 + 0.988i)T \)
11 \( 1 + (0.883 - 0.467i)T \)
13 \( 1 + (-0.999 - 0.0442i)T \)
17 \( 1 + (0.862 + 0.506i)T \)
19 \( 1 + (-0.699 - 0.714i)T \)
23 \( 1 + (-0.984 + 0.176i)T \)
29 \( 1 + (-0.862 + 0.506i)T \)
31 \( 1 + (0.730 - 0.683i)T \)
37 \( 1 + (-0.154 - 0.988i)T \)
41 \( 1 + (0.984 - 0.176i)T \)
43 \( 1 + (-0.921 - 0.387i)T \)
47 \( 1 + (-0.964 + 0.262i)T \)
53 \( 1 + (-0.0663 - 0.997i)T \)
59 \( 1 + (-0.964 + 0.262i)T \)
61 \( 1 + (0.814 - 0.580i)T \)
67 \( 1 + (0.862 - 0.506i)T \)
71 \( 1 + (-0.367 - 0.930i)T \)
73 \( 1 + (-0.699 - 0.714i)T \)
79 \( 1 + (-0.991 - 0.132i)T \)
83 \( 1 + (0.952 - 0.304i)T \)
89 \( 1 + (0.730 + 0.683i)T \)
97 \( 1 + (0.110 - 0.993i)T \)
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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

−23.26397627964866313079433237161, −22.70519941782438277937075644420, −21.74789603313257978473761687674, −20.40487898822319762606865708084, −19.81627644964713027973700036011, −19.14311554099510748808164456864, −18.49749055493584904979606723087, −17.52114038711971895347920677016, −16.989424419944882872688158461269, −16.200759941987541396103760572254, −14.76629174749477123861994736961, −14.14627830104487182920240388443, −12.82401601516404241439998877291, −11.9103031657474962132695413296, −11.55919452338014612796812145464, −10.41162252380245319123595819184, −9.81720996845588576600339304619, −8.26672867040146331357250689264, −7.62861451936301828657119138433, −7.00093003958357132528619198026, −6.254033425852795912330557431550, −4.41621775941023818662344550757, −3.33127671324904141159749938775, −2.18565634066111615513230265305, −1.10985783211614986087083828428, 0.30904968696893145079837347414, 1.93189385041616208385685882743, 3.38580611062949236449259761167, 4.628775743090705516557977567206, 5.50958830787021617090755648400, 6.29799719931445357080540341036, 7.711244440400610656519976298668, 8.58238693778924660279463829251, 9.21144318742801268802648050574, 9.93915674366335665585164830748, 11.161728275775424330120217161548, 11.76124994766490090233908404405, 12.50561543021638925637248015096, 14.42396336193709583266300404946, 14.97729726675497833071345292864, 15.73055939378644054537163294035, 16.52719754827702949015878271090, 17.065608395206395371384870521657, 17.898232942046659036121610445552, 19.22452047261863098046380736957, 19.57204852642494999439300671121, 20.587391178750653544759579605347, 21.43574483485484671363310151880, 22.20061640395671886229611891231, 23.31230066722555877379618551520

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