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Properties

Label 1-1004-1004.167-r0-0-0
Degree $1$
Conductor $1004$
Sign $0.995 + 0.0903i$
Analytic cond. $4.66255$
Root an. cond. $4.66255$
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  + (−0.988 + 0.150i)3-s + (−0.425 − 0.904i)5-s + (−0.693 − 0.720i)7-s + (0.954 − 0.297i)9-s + (−0.999 + 0.0251i)11-s + (−0.597 + 0.801i)13-s + (0.556 + 0.830i)15-s + (−0.0878 + 0.996i)17-s + (−0.910 − 0.414i)19-s + (0.793 + 0.607i)21-s + (0.974 − 0.224i)23-s + (−0.637 + 0.770i)25-s + (−0.899 + 0.437i)27-s + (0.379 + 0.925i)29-s + (−0.823 − 0.567i)31-s + ⋯
L(s)  = 1  + (−0.988 + 0.150i)3-s + (−0.425 − 0.904i)5-s + (−0.693 − 0.720i)7-s + (0.954 − 0.297i)9-s + (−0.999 + 0.0251i)11-s + (−0.597 + 0.801i)13-s + (0.556 + 0.830i)15-s + (−0.0878 + 0.996i)17-s + (−0.910 − 0.414i)19-s + (0.793 + 0.607i)21-s + (0.974 − 0.224i)23-s + (−0.637 + 0.770i)25-s + (−0.899 + 0.437i)27-s + (0.379 + 0.925i)29-s + (−0.823 − 0.567i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1004\)    =    \(2^{2} \cdot 251\)
Sign: $0.995 + 0.0903i$
Analytic conductor: \(4.66255\)
Root analytic conductor: \(4.66255\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1004} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1004,\ (0:\ ),\ 0.995 + 0.0903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5166863280 + 0.02338060782i\)
\(L(\frac12)\) \(\approx\) \(0.5166863280 + 0.02338060782i\)
\(L(1)\) \(\approx\) \(0.5681223312 - 0.06084729472i\)
\(L(1)\) \(\approx\) \(0.5681223312 - 0.06084729472i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 \)
good3 \( 1 + (-0.988 + 0.150i)T \)
5 \( 1 + (-0.425 - 0.904i)T \)
7 \( 1 + (-0.693 - 0.720i)T \)
11 \( 1 + (-0.999 + 0.0251i)T \)
13 \( 1 + (-0.597 + 0.801i)T \)
17 \( 1 + (-0.0878 + 0.996i)T \)
19 \( 1 + (-0.910 - 0.414i)T \)
23 \( 1 + (0.974 - 0.224i)T \)
29 \( 1 + (0.379 + 0.925i)T \)
31 \( 1 + (-0.823 - 0.567i)T \)
37 \( 1 + (-0.0125 + 0.999i)T \)
41 \( 1 + (-0.0376 - 0.999i)T \)
43 \( 1 + (0.356 + 0.934i)T \)
47 \( 1 + (0.535 + 0.844i)T \)
53 \( 1 + (0.837 - 0.546i)T \)
59 \( 1 + (-0.0878 - 0.996i)T \)
61 \( 1 + (-0.260 - 0.965i)T \)
67 \( 1 + (0.556 - 0.830i)T \)
71 \( 1 + (-0.947 - 0.320i)T \)
73 \( 1 + (0.448 - 0.893i)T \)
79 \( 1 + (-0.762 - 0.647i)T \)
83 \( 1 + (-0.994 + 0.100i)T \)
89 \( 1 + (0.656 - 0.754i)T \)
97 \( 1 + (0.863 + 0.503i)T \)
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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

−21.7403941239002358388371963608, −21.18011993954726743783242120079, −19.866841156925286362598636329111, −19.05455880097947325057575597146, −18.45704663926869629697732823546, −17.926717125822162373676510741367, −16.94955160318674716422609623782, −16.00894692358666998731969813972, −15.46631385976060023657117592249, −14.803716524482346797576557961031, −13.481698675543227077415188740555, −12.73153560679599672986322337845, −12.066053988351794062517084529338, −11.2224032654103411670637513365, −10.432716550627669170139342123607, −9.89307210023424097372840515921, −8.617869356995086685790532500113, −7.414875261521563847848850622616, −7.01821234079547548985731679753, −5.867565415774966552195152036829, −5.370582983429903803891103866719, −4.202265414395866979863146280122, −2.95054783467861446685044542822, −2.31990783656890141422628969719, −0.438258282144847319589881209297, 0.62329625080424702782702102596, 1.873662633310139323644517248547, 3.4295388319420431794935409633, 4.44937472472704861080495300882, 4.90036041119560587974668431725, 6.01600332234503446494606892834, 6.885397970080807630553745596288, 7.65526200122280536142923734851, 8.80012487576475305246412926329, 9.6483822502055338018610582366, 10.586094943386934115275065558434, 11.109465861988848235744336710822, 12.26855776841489150297489744611, 12.81853471224256058655761944106, 13.31816969460884376567708575203, 14.734351442127543537625825830618, 15.6498317583971075023113690585, 16.254283465331538448645800028075, 17.04494433936359918716047735678, 17.29431618147649674078987795930, 18.64549346572663105977370924512, 19.2622814084425710856304173392, 20.108517656209223135856922803135, 21.016667729753480512913325166241, 21.608301694912011246771389726980

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