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Properties

Label 1-4553-4553.1024-r0-0-0
Degree $1$
Conductor $4553$
Sign $-0.709 - 0.704i$
Analytic cond. $21.1440$
Root an. cond. $21.1440$
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  + (0.188 − 0.982i)2-s + (0.928 − 0.370i)3-s + (−0.928 − 0.370i)4-s + (0.792 − 0.609i)5-s + (−0.188 − 0.982i)6-s + (0.928 − 0.370i)7-s + (−0.539 + 0.842i)8-s + (0.725 − 0.688i)9-s + (−0.449 − 0.893i)10-s + (0.999 + 0.0345i)11-s − 12-s + (−0.900 + 0.433i)13-s + (−0.188 − 0.982i)14-s + (0.509 − 0.860i)15-s + (0.725 + 0.688i)16-s + (0.970 + 0.239i)17-s + ⋯
L(s)  = 1  + (0.188 − 0.982i)2-s + (0.928 − 0.370i)3-s + (−0.928 − 0.370i)4-s + (0.792 − 0.609i)5-s + (−0.188 − 0.982i)6-s + (0.928 − 0.370i)7-s + (−0.539 + 0.842i)8-s + (0.725 − 0.688i)9-s + (−0.449 − 0.893i)10-s + (0.999 + 0.0345i)11-s − 12-s + (−0.900 + 0.433i)13-s + (−0.188 − 0.982i)14-s + (0.509 − 0.860i)15-s + (0.725 + 0.688i)16-s + (0.970 + 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4553 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4553 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4553\)    =    \(29 \cdot 157\)
Sign: $-0.709 - 0.704i$
Analytic conductor: \(21.1440\)
Root analytic conductor: \(21.1440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4553} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4553,\ (0:\ ),\ -0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.480778389 - 3.594479430i\)
\(L(\frac12)\) \(\approx\) \(1.480778389 - 3.594479430i\)
\(L(1)\) \(\approx\) \(1.418698338 - 1.410749954i\)
\(L(1)\) \(\approx\) \(1.418698338 - 1.410749954i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
157 \( 1 \)
good2 \( 1 + (0.188 - 0.982i)T \)
3 \( 1 + (0.928 - 0.370i)T \)
5 \( 1 + (0.792 - 0.609i)T \)
7 \( 1 + (0.928 - 0.370i)T \)
11 \( 1 + (0.999 + 0.0345i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.970 + 0.239i)T \)
19 \( 1 + (0.994 - 0.103i)T \)
23 \( 1 + (-0.509 + 0.860i)T \)
31 \( 1 + (-0.0517 - 0.998i)T \)
37 \( 1 + (-0.978 - 0.205i)T \)
41 \( 1 + (-0.354 + 0.935i)T \)
43 \( 1 + (0.997 - 0.0689i)T \)
47 \( 1 + (-0.978 + 0.205i)T \)
53 \( 1 + (0.700 + 0.713i)T \)
59 \( 1 + (-0.120 - 0.992i)T \)
61 \( 1 + (0.940 + 0.338i)T \)
67 \( 1 + (-0.539 + 0.842i)T \)
71 \( 1 + (0.322 + 0.946i)T \)
73 \( 1 + (-0.289 + 0.957i)T \)
79 \( 1 + (-0.596 - 0.802i)T \)
83 \( 1 + (-0.813 - 0.582i)T \)
89 \( 1 + (-0.0517 - 0.998i)T \)
97 \( 1 + (-0.0172 + 0.999i)T \)
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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

−18.203719535092885633084992585909, −17.8947482751266837874902057959, −17.037938115713075505247631132536, −16.47143520079058385490964676431, −15.60651243884352776863028882338, −14.899657381807777581808808341802, −14.44083120406756023892865383850, −14.11407650825166793742520061165, −13.58991334029713664651147873589, −12.43122483443807805386534540369, −12.01104911453829266661664160544, −10.75983564807489796964851696418, −9.96631579674978972562685726866, −9.54804182066872423478802912496, −8.74597969740756343428138861134, −8.19930017152325911446440006016, −7.31348679570041350626479233323, −6.97253950482964617463737598249, −5.84183422339670621887112581220, −5.23604110268714823415452154919, −4.636358616484323373141620107050, −3.61517892497554553143415036463, −3.03488868985408496853244659152, −2.10729986747816376715541790303, −1.20702977151014950705313982800, 0.97268923717194494759246885902, 1.47982650687409059948593103769, 2.03025820794038113616178365883, 2.90277675298677324558079901919, 3.83344940625081614164655186213, 4.37969620863340523300399820968, 5.22317127921427335561572276589, 5.9115597071842209739954486804, 7.07297519501417880993561295176, 7.81233694867709618565971660400, 8.52794856872639177716955232884, 9.25198707558894710867183809607, 9.76300138960549425854920402289, 10.19657221211422570127678156676, 11.51414998819200794447815536406, 11.860126907901502406022130163006, 12.57751869640582722494247117099, 13.33440677939764713599708893516, 13.901387487722407225859300825572, 14.494278132707342276112609243965, 14.65388049684868163504166992461, 15.9262053249335275406033626087, 17.05725321236772739030252820431, 17.38879116813101310275204175461, 18.064689390089443616154256116601

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