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Properties

Label 1-569-569.18-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.148 - 0.988i$
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  + (−0.197 − 0.980i)2-s + (0.814 − 0.580i)3-s + (−0.921 + 0.387i)4-s + (0.408 − 0.912i)5-s + (−0.730 − 0.683i)6-s + (0.759 + 0.650i)7-s + (0.562 + 0.826i)8-s + (0.325 − 0.945i)9-s + (−0.975 − 0.219i)10-s + (0.240 + 0.970i)11-s + (−0.525 + 0.850i)12-s + (−0.0221 + 0.999i)13-s + (0.487 − 0.873i)14-s + (−0.197 − 0.980i)15-s + (0.699 − 0.714i)16-s + (0.964 + 0.262i)17-s + ⋯
L(s)  = 1  + (−0.197 − 0.980i)2-s + (0.814 − 0.580i)3-s + (−0.921 + 0.387i)4-s + (0.408 − 0.912i)5-s + (−0.730 − 0.683i)6-s + (0.759 + 0.650i)7-s + (0.562 + 0.826i)8-s + (0.325 − 0.945i)9-s + (−0.975 − 0.219i)10-s + (0.240 + 0.970i)11-s + (−0.525 + 0.850i)12-s + (−0.0221 + 0.999i)13-s + (0.487 − 0.873i)14-s + (−0.197 − 0.980i)15-s + (0.699 − 0.714i)16-s + (0.964 + 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.148 - 0.988i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.148 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.216931915 - 1.413102054i\)
\(L(\frac12)\) \(\approx\) \(1.216931915 - 1.413102054i\)
\(L(1)\) \(\approx\) \(1.111928776 - 0.8232632279i\)
\(L(1)\) \(\approx\) \(1.111928776 - 0.8232632279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.197 - 0.980i)T \)
3 \( 1 + (0.814 - 0.580i)T \)
5 \( 1 + (0.408 - 0.912i)T \)
7 \( 1 + (0.759 + 0.650i)T \)
11 \( 1 + (0.240 + 0.970i)T \)
13 \( 1 + (-0.0221 + 0.999i)T \)
17 \( 1 + (0.964 + 0.262i)T \)
19 \( 1 + (-0.921 - 0.387i)T \)
23 \( 1 + (0.996 - 0.0883i)T \)
29 \( 1 + (0.964 - 0.262i)T \)
31 \( 1 + (-0.367 - 0.930i)T \)
37 \( 1 + (0.759 + 0.650i)T \)
41 \( 1 + (0.996 - 0.0883i)T \)
43 \( 1 + (-0.197 + 0.980i)T \)
47 \( 1 + (-0.991 + 0.132i)T \)
53 \( 1 + (-0.730 - 0.683i)T \)
59 \( 1 + (-0.991 + 0.132i)T \)
61 \( 1 + (-0.952 + 0.304i)T \)
67 \( 1 + (0.964 - 0.262i)T \)
71 \( 1 + (0.562 - 0.826i)T \)
73 \( 1 + (-0.921 - 0.387i)T \)
79 \( 1 + (0.0663 - 0.997i)T \)
83 \( 1 + (0.154 + 0.988i)T \)
89 \( 1 + (-0.367 + 0.930i)T \)
97 \( 1 + (-0.666 - 0.745i)T \)
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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

−23.381826778498215101631952460, −22.89845340358001805584888593739, −21.62594449733263550405713950395, −21.35129170576932225642125815025, −20.04883021387142570769191107654, −19.150653916298460144296778449567, −18.46318404883128001035183946635, −17.47551617203200960098201545358, −16.75102955810721702760529269901, −15.81523075249419778145237346113, −14.862962319564032477678385660518, −14.35412653303170127518535636157, −13.848457851593683941373072932696, −12.84971973881869819121415127584, −10.90506607763905937771989213693, −10.52057473446623099698864907009, −9.57446304848908639855235320014, −8.518251384919243250166541388142, −7.85230619668425135940361486487, −7.02458452066398230580352214381, −5.84520804243029106887157608140, −4.95095964872280183886101089923, −3.780506481913707706938168755299, −2.94257540672767943953659052313, −1.28693588388526796043756367001, 1.25607597060872911648724102264, 1.88069684112995969390412210824, 2.75294843714403200147395094191, 4.25271950435304217129359695137, 4.833169544956715454665881239192, 6.28957916745147113711748731846, 7.734625705534956561780560272, 8.41906187959334164446786292802, 9.278770701032096533576743573677, 9.7087954394192110777604001515, 11.20253221240688913278032551013, 12.168866748564078630226717771879, 12.62054962030268017822043280657, 13.47449320661390012310182783703, 14.42315807368856748130174800770, 15.0425793563182407721849576875, 16.65446037442135398977438581927, 17.46116346567271647472873170491, 18.15541578610313360424653559981, 19.0670075255746052560224517201, 19.66737216574456185991916424921, 20.67236542300965806762297408480, 21.116833918454866378744092276311, 21.67794069913853040949479346399, 23.148824884717461303778693556891

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