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Properties

Label 1-569-569.182-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.664 + 0.747i$
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  + (−0.699 + 0.714i)2-s + (−0.616 + 0.787i)3-s + (−0.0221 − 0.999i)4-s + (−0.110 + 0.993i)5-s + (−0.132 − 0.991i)6-s + (0.952 − 0.304i)7-s + (0.730 + 0.683i)8-s + (−0.240 − 0.970i)9-s + (−0.633 − 0.773i)10-s + (0.826 + 0.562i)11-s + (0.801 + 0.598i)12-s + (−0.996 − 0.0883i)13-s + (−0.448 + 0.894i)14-s + (−0.714 − 0.699i)15-s + (−0.999 + 0.0442i)16-s + (−0.487 − 0.873i)17-s + ⋯
L(s)  = 1  + (−0.699 + 0.714i)2-s + (−0.616 + 0.787i)3-s + (−0.0221 − 0.999i)4-s + (−0.110 + 0.993i)5-s + (−0.132 − 0.991i)6-s + (0.952 − 0.304i)7-s + (0.730 + 0.683i)8-s + (−0.240 − 0.970i)9-s + (−0.633 − 0.773i)10-s + (0.826 + 0.562i)11-s + (0.801 + 0.598i)12-s + (−0.996 − 0.0883i)13-s + (−0.448 + 0.894i)14-s + (−0.714 − 0.699i)15-s + (−0.999 + 0.0442i)16-s + (−0.487 − 0.873i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.664 + 0.747i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.664 + 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3408249333 + 0.7585276965i\)
\(L(\frac12)\) \(\approx\) \(0.3408249333 + 0.7585276965i\)
\(L(1)\) \(\approx\) \(0.5497395222 + 0.4542736371i\)
\(L(1)\) \(\approx\) \(0.5497395222 + 0.4542736371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.699 + 0.714i)T \)
3 \( 1 + (-0.616 + 0.787i)T \)
5 \( 1 + (-0.110 + 0.993i)T \)
7 \( 1 + (0.952 - 0.304i)T \)
11 \( 1 + (0.826 + 0.562i)T \)
13 \( 1 + (-0.996 - 0.0883i)T \)
17 \( 1 + (-0.487 - 0.873i)T \)
19 \( 1 + (0.999 + 0.0221i)T \)
23 \( 1 + (0.346 + 0.937i)T \)
29 \( 1 + (0.873 + 0.487i)T \)
31 \( 1 + (0.997 + 0.0663i)T \)
37 \( 1 + (-0.304 - 0.952i)T \)
41 \( 1 + (0.937 - 0.346i)T \)
43 \( 1 + (0.699 + 0.714i)T \)
47 \( 1 + (0.506 + 0.862i)T \)
53 \( 1 + (0.132 + 0.991i)T \)
59 \( 1 + (-0.506 - 0.862i)T \)
61 \( 1 + (-0.325 + 0.945i)T \)
67 \( 1 + (-0.487 + 0.873i)T \)
71 \( 1 + (0.730 - 0.683i)T \)
73 \( 1 + (-0.999 - 0.0221i)T \)
79 \( 1 + (-0.964 - 0.262i)T \)
83 \( 1 + (0.580 + 0.814i)T \)
89 \( 1 + (-0.997 + 0.0663i)T \)
97 \( 1 + (0.219 - 0.975i)T \)
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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

−22.81305860280237377639841254540, −21.94609372785277811439413648455, −21.2787263515265477846102909338, −20.24525586200554254502981481146, −19.54823568764833645930002293024, −18.86907829959350623583492394426, −17.8424560488523679906743519915, −17.18828623867173845024164648557, −16.79306556605591491779475830764, −15.67119400598648658378076461937, −14.19017355292248182674664439228, −13.32175743192326724124635488128, −12.25190633737234277179825932129, −11.94918110948978499310138618497, −11.169519308790413721902525991765, −10.094447457684177508727001897568, −8.86750161175949867082076617637, −8.30685565063396643909525530986, −7.467597863603849465961642888687, −6.30265274945205105247142670544, −5.02764348129911861699716282677, −4.2720346877041453317433724649, −2.58198585475305714603571089452, −1.56747511383933705269473280914, −0.752760337870289066179395588718, 1.13108153277872683754171166841, 2.69088281519985001181164373064, 4.261837002343434282459358198910, 4.99121310714224654662873534009, 6.01939326187477405930589481181, 7.13295391600917770611002592852, 7.524312848269546727467862849658, 9.05654765945564510597752821455, 9.72678207651065813823110048097, 10.56425617614995158357625513707, 11.33940475753523891417494047928, 11.9901470471536868586305564897, 14.10021282895578749236713189451, 14.361686982638003682907981045942, 15.33492063853536439698851727124, 15.8917424579063080543843005498, 17.068160254850974814263218342153, 17.71156282131315057038642017564, 18.01587912152038997845372050399, 19.37476356375263226927564120062, 20.06656885380231947827483501277, 21.10858157667889864730266928903, 22.21036243185867563478022006556, 22.74321196952491843714513500443, 23.49110569108273057906284452566

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