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Properties

Label 1-1004-1004.263-r1-0-0
Degree $1$
Conductor $1004$
Sign $0.999 + 0.00750i$
Analytic cond. $107.894$
Root an. cond. $107.894$
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  + (−0.793 − 0.607i)3-s + (−0.187 − 0.982i)5-s + (−0.762 − 0.647i)7-s + (0.260 + 0.965i)9-s + (−0.402 − 0.915i)11-s + (0.356 − 0.934i)13-s + (−0.448 + 0.893i)15-s + (0.617 − 0.786i)17-s + (−0.693 + 0.720i)19-s + (0.212 + 0.977i)21-s + (0.556 + 0.830i)23-s + (−0.929 + 0.368i)25-s + (0.379 − 0.925i)27-s + (−0.597 + 0.801i)29-s + (0.863 + 0.503i)31-s + ⋯
L(s)  = 1  + (−0.793 − 0.607i)3-s + (−0.187 − 0.982i)5-s + (−0.762 − 0.647i)7-s + (0.260 + 0.965i)9-s + (−0.402 − 0.915i)11-s + (0.356 − 0.934i)13-s + (−0.448 + 0.893i)15-s + (0.617 − 0.786i)17-s + (−0.693 + 0.720i)19-s + (0.212 + 0.977i)21-s + (0.556 + 0.830i)23-s + (−0.929 + 0.368i)25-s + (0.379 − 0.925i)27-s + (−0.597 + 0.801i)29-s + (0.863 + 0.503i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1004\)    =    \(2^{2} \cdot 251\)
Sign: $0.999 + 0.00750i$
Analytic conductor: \(107.894\)
Root analytic conductor: \(107.894\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1004} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1004,\ (1:\ ),\ 0.999 + 0.00750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5674710483 + 0.002129253800i\)
\(L(\frac12)\) \(\approx\) \(0.5674710483 + 0.002129253800i\)
\(L(1)\) \(\approx\) \(0.5893773544 - 0.2923666089i\)
\(L(1)\) \(\approx\) \(0.5893773544 - 0.2923666089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 \)
good3 \( 1 + (-0.793 - 0.607i)T \)
5 \( 1 + (-0.187 - 0.982i)T \)
7 \( 1 + (-0.762 - 0.647i)T \)
11 \( 1 + (-0.402 - 0.915i)T \)
13 \( 1 + (0.356 - 0.934i)T \)
17 \( 1 + (0.617 - 0.786i)T \)
19 \( 1 + (-0.693 + 0.720i)T \)
23 \( 1 + (0.556 + 0.830i)T \)
29 \( 1 + (-0.597 + 0.801i)T \)
31 \( 1 + (0.863 + 0.503i)T \)
37 \( 1 + (-0.837 + 0.546i)T \)
41 \( 1 + (0.162 - 0.986i)T \)
43 \( 1 + (-0.492 - 0.870i)T \)
47 \( 1 + (0.637 + 0.770i)T \)
53 \( 1 + (0.112 + 0.993i)T \)
59 \( 1 + (-0.617 - 0.786i)T \)
61 \( 1 + (-0.910 + 0.414i)T \)
67 \( 1 + (-0.448 - 0.893i)T \)
71 \( 1 + (0.778 + 0.627i)T \)
73 \( 1 + (0.823 + 0.567i)T \)
79 \( 1 + (-0.577 + 0.816i)T \)
83 \( 1 + (0.0878 + 0.996i)T \)
89 \( 1 + (-0.0376 + 0.999i)T \)
97 \( 1 + (0.656 + 0.754i)T \)
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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

−21.41597655713701920156369363273, −21.10724133798138456769688252915, −19.766238016045127918073008834105, −18.910784789249544711073711415614, −18.425575492863427463390912798178, −17.492399819127207478712710326304, −16.7266178059022395020752582365, −15.882532507549557898358571389660, −15.08472180085750457906682834612, −14.81350747119655202198648483857, −13.40678101720987726957090322726, −12.51255548376630215954091174606, −11.78986978548949231967817741146, −10.99178143841310028145724646482, −10.208093871348478993309870634793, −9.60657481770998261130506363916, −8.63051070190245764047445466030, −7.31634072720820560441298233847, −6.45844423311714942998790780796, −6.03798272500010556787557123597, −4.78188830977507716165627936498, −3.976363389163286021530213142686, −2.98988978035641449164179341110, −1.97344687811175219777094255521, −0.204521429549507119670133791990, 0.692918187094835288377184441, 1.36705030714407829698467253926, 2.97741195928927712022174228295, 3.90997670331458355543238799254, 5.20698142275620679982515056148, 5.62090781925954518500814599706, 6.656067134774701304715661680117, 7.612048318239277607762700388570, 8.26633778518306937975300068291, 9.32811358691679381625321885602, 10.42989823845697851217786826279, 10.95251789875590355585106125995, 12.15687066440585168474764596490, 12.56598084601082022699089971740, 13.48157329905345838631919929162, 13.84558860690985492733704914376, 15.57430019146996939443475887894, 16.034391045684914818226264364422, 16.95177067960633885965644198084, 17.18497090560552338817400489380, 18.45536771997072606862531465373, 19.01637915463881566432483156460, 19.825202411200172415776607919464, 20.65609720329501456106005287051, 21.39920838123296651503182702788

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