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Properties

Label 1-569-569.20-r0-0-0
Degree $1$
Conductor $569$
Sign $0.640 + 0.768i$
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  + (−0.730 − 0.683i)2-s + (−0.408 + 0.912i)3-s + (0.0663 + 0.997i)4-s + (0.325 − 0.945i)5-s + (0.921 − 0.387i)6-s + (−0.598 + 0.801i)7-s + (0.633 − 0.773i)8-s + (−0.666 − 0.745i)9-s + (−0.883 + 0.467i)10-s + (0.975 + 0.219i)11-s + (−0.937 − 0.346i)12-s + (0.964 + 0.262i)13-s + (0.984 − 0.176i)14-s + (0.730 + 0.683i)15-s + (−0.991 + 0.132i)16-s + (−0.999 − 0.0442i)17-s + ⋯
L(s)  = 1  + (−0.730 − 0.683i)2-s + (−0.408 + 0.912i)3-s + (0.0663 + 0.997i)4-s + (0.325 − 0.945i)5-s + (0.921 − 0.387i)6-s + (−0.598 + 0.801i)7-s + (0.633 − 0.773i)8-s + (−0.666 − 0.745i)9-s + (−0.883 + 0.467i)10-s + (0.975 + 0.219i)11-s + (−0.937 − 0.346i)12-s + (0.964 + 0.262i)13-s + (0.984 − 0.176i)14-s + (0.730 + 0.683i)15-s + (−0.991 + 0.132i)16-s + (−0.999 − 0.0442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.640 + 0.768i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.640 + 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6778959525 + 0.3176014369i\)
\(L(\frac12)\) \(\approx\) \(0.6778959525 + 0.3176014369i\)
\(L(1)\) \(\approx\) \(0.6783993325 + 0.04991298465i\)
\(L(1)\) \(\approx\) \(0.6783993325 + 0.04991298465i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.730 - 0.683i)T \)
3 \( 1 + (-0.408 + 0.912i)T \)
5 \( 1 + (0.325 - 0.945i)T \)
7 \( 1 + (-0.598 + 0.801i)T \)
11 \( 1 + (0.975 + 0.219i)T \)
13 \( 1 + (0.964 + 0.262i)T \)
17 \( 1 + (-0.999 - 0.0442i)T \)
19 \( 1 + (-0.0663 + 0.997i)T \)
23 \( 1 + (-0.487 + 0.873i)T \)
29 \( 1 + (0.999 - 0.0442i)T \)
31 \( 1 + (0.197 - 0.980i)T \)
37 \( 1 + (0.598 - 0.801i)T \)
41 \( 1 + (0.487 - 0.873i)T \)
43 \( 1 + (-0.730 + 0.683i)T \)
47 \( 1 + (0.0221 + 0.999i)T \)
53 \( 1 + (0.921 - 0.387i)T \)
59 \( 1 + (0.0221 + 0.999i)T \)
61 \( 1 + (-0.839 + 0.544i)T \)
67 \( 1 + (-0.999 + 0.0442i)T \)
71 \( 1 + (0.633 + 0.773i)T \)
73 \( 1 + (-0.0663 + 0.997i)T \)
79 \( 1 + (0.699 + 0.714i)T \)
83 \( 1 + (0.283 + 0.958i)T \)
89 \( 1 + (0.197 + 0.980i)T \)
97 \( 1 + (0.787 + 0.616i)T \)
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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

−23.24349636607043558222416281295, −22.64440560640260156737191688352, −21.837278838035239059033282177499, −19.98569251886152140037468941196, −19.73456691370365820622459460183, −18.72329879640288667838517756410, −18.04415552705026954055601975959, −17.43337373918688784839907350012, −16.64110696248796378401961022802, −15.75494821379593749620228261908, −14.65884566678557193983477939110, −13.70874672465101728849274232297, −13.38876480847707040721387135398, −11.79761179307612685411182426287, −10.8613985749029379204347605229, −10.38810148186336620564369283095, −9.11740211559484769328674569229, −8.223034330836365554927367715929, −7.08171067883730564503011742658, −6.47462581626508670913960973241, −6.20881718319437740168262639220, −4.630958351288498654246956184847, −3.0696825857799336759921830435, −1.790309817082906272054966964774, −0.62248382382359769370339139864, 1.122991767703118100979173538978, 2.35034578020695956714015842601, 3.75945096171795182436191751967, 4.318403077910886262608829087165, 5.74550222745820291471793102649, 6.47520340256366621342952881599, 8.20079734978852775536365526260, 9.06481852244186674806304296850, 9.39558639189832705182556398249, 10.29307442420754239833002158450, 11.46006032197283082353063245526, 11.98550625109851002277603659272, 12.85322899833703994661258949768, 13.8837752758191091385896090753, 15.340954865850751061800834883355, 16.125702371551976909601675091485, 16.6200195874900308891538233826, 17.5565236046963208269758347338, 18.187907046320772005631835859405, 19.46199820220114794884543578601, 20.03337726523628319025087838477, 21.00672099380872904046105068478, 21.45473260381426386511436376364, 22.30614960940336920152677149091, 22.9945322736605222986723223969

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