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Properties

Label 1-1004-1004.79-r1-0-0
Degree $1$
Conductor $1004$
Sign $-0.222 + 0.975i$
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  + (−0.762 + 0.647i)3-s + (0.535 − 0.844i)5-s + (0.0878 − 0.996i)7-s + (0.162 − 0.986i)9-s + (0.597 + 0.801i)11-s + (0.988 − 0.150i)13-s + (0.137 + 0.990i)15-s + (0.112 + 0.993i)17-s + (−0.994 + 0.100i)19-s + (0.577 + 0.816i)21-s + (−0.492 + 0.870i)23-s + (−0.425 − 0.904i)25-s + (0.514 + 0.857i)27-s + (−0.962 − 0.272i)29-s + (0.947 − 0.320i)31-s + ⋯
L(s)  = 1  + (−0.762 + 0.647i)3-s + (0.535 − 0.844i)5-s + (0.0878 − 0.996i)7-s + (0.162 − 0.986i)9-s + (0.597 + 0.801i)11-s + (0.988 − 0.150i)13-s + (0.137 + 0.990i)15-s + (0.112 + 0.993i)17-s + (−0.994 + 0.100i)19-s + (0.577 + 0.816i)21-s + (−0.492 + 0.870i)23-s + (−0.425 − 0.904i)25-s + (0.514 + 0.857i)27-s + (−0.962 − 0.272i)29-s + (0.947 − 0.320i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1004\)    =    \(2^{2} \cdot 251\)
Sign: $-0.222 + 0.975i$
Analytic conductor: \(107.894\)
Root analytic conductor: \(107.894\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1004} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1004,\ (1:\ ),\ -0.222 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6644365567 + 0.8327419426i\)
\(L(\frac12)\) \(\approx\) \(0.6644365567 + 0.8327419426i\)
\(L(1)\) \(\approx\) \(0.8882489196 + 0.07666393353i\)
\(L(1)\) \(\approx\) \(0.8882489196 + 0.07666393353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 \)
good3 \( 1 + (-0.762 + 0.647i)T \)
5 \( 1 + (0.535 - 0.844i)T \)
7 \( 1 + (0.0878 - 0.996i)T \)
11 \( 1 + (0.597 + 0.801i)T \)
13 \( 1 + (0.988 - 0.150i)T \)
17 \( 1 + (0.112 + 0.993i)T \)
19 \( 1 + (-0.994 + 0.100i)T \)
23 \( 1 + (-0.492 + 0.870i)T \)
29 \( 1 + (-0.962 - 0.272i)T \)
31 \( 1 + (0.947 - 0.320i)T \)
37 \( 1 + (0.448 + 0.893i)T \)
41 \( 1 + (-0.984 + 0.175i)T \)
43 \( 1 + (-0.793 + 0.607i)T \)
47 \( 1 + (-0.876 - 0.481i)T \)
53 \( 1 + (0.823 - 0.567i)T \)
59 \( 1 + (-0.112 + 0.993i)T \)
61 \( 1 + (-0.332 + 0.942i)T \)
67 \( 1 + (0.137 - 0.990i)T \)
71 \( 1 + (0.888 + 0.459i)T \)
73 \( 1 + (-0.997 - 0.0753i)T \)
79 \( 1 + (-0.617 + 0.786i)T \)
83 \( 1 + (0.837 + 0.546i)T \)
89 \( 1 + (0.979 + 0.199i)T \)
97 \( 1 + (-0.778 + 0.627i)T \)
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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

−21.62711496559301139409626469419, −20.55724546166178615448141366119, −19.23247374745946547225227892626, −18.69222781596663519710541532599, −18.282992275048892823363292075163, −17.46968919586462389824949872992, −16.58031086212229742952560771446, −15.85682478597257823971301258800, −14.80756098220411996986321676371, −13.99342574135251949053057105520, −13.32619552680084637730506700444, −12.37390946551593550155942983765, −11.48813407522407203883419362332, −11.06389006529618062139471124976, −10.11683872320325852980900212944, −8.98147253427089968401395659662, −8.25348406391491286752497791233, −7.03580513795923427463398542345, −6.26259660167036806662842718165, −5.895949843652773125803192564851, −4.84339196628516780549196295487, −3.45002395723200044681466332898, −2.40124909855477227534408698524, −1.60633463070238503097957947994, −0.265534420926400516479479702784, 1.05397072056729970705729550776, 1.737081456774538044116088388434, 3.68383372655767177538439771630, 4.19530375577519906224622128499, 5.02728138584849904496572263456, 6.083001043216575399787828454929, 6.60073815329839301720926028804, 7.95820264873171167092718231098, 8.821224743464695282327242930648, 9.90143821483111746089377178171, 10.19805558128382065085634660522, 11.26462167810823546243735244048, 12.00473279867586392648593279032, 13.03172567333740043195845741660, 13.50811794190206761942431370029, 14.77156509625336066235025317963, 15.38179008931103723938277995131, 16.52556891390239506102875262300, 16.90973571065035435776830809392, 17.48806171982807647315412322709, 18.23791512263203964297121356645, 19.60547229618271860124685134990, 20.27000389940722307332713361928, 21.00547582216850753052467128523, 21.501626217136619621047729540311

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