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Properties

Label 1-2004-2004.755-r0-0-0
Degree $1$
Conductor $2004$
Sign $0.997 - 0.0743i$
Analytic cond. $9.30653$
Root an. cond. $9.30653$
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  + (0.982 − 0.188i)5-s + (0.942 − 0.334i)7-s + (0.553 + 0.832i)11-s + (−0.800 + 0.599i)13-s + (−0.822 + 0.569i)17-s + (0.0189 − 0.999i)19-s + (−0.993 − 0.113i)23-s + (0.929 − 0.369i)25-s + (0.993 − 0.113i)29-s + (0.243 − 0.969i)31-s + (0.862 − 0.505i)35-s + (−0.521 − 0.853i)37-s + (0.881 + 0.472i)41-s + (−0.726 + 0.686i)43-s + (0.988 + 0.150i)47-s + ⋯
L(s)  = 1  + (0.982 − 0.188i)5-s + (0.942 − 0.334i)7-s + (0.553 + 0.832i)11-s + (−0.800 + 0.599i)13-s + (−0.822 + 0.569i)17-s + (0.0189 − 0.999i)19-s + (−0.993 − 0.113i)23-s + (0.929 − 0.369i)25-s + (0.993 − 0.113i)29-s + (0.243 − 0.969i)31-s + (0.862 − 0.505i)35-s + (−0.521 − 0.853i)37-s + (0.881 + 0.472i)41-s + (−0.726 + 0.686i)43-s + (0.988 + 0.150i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
Sign: $0.997 - 0.0743i$
Analytic conductor: \(9.30653\)
Root analytic conductor: \(9.30653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2004} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2004,\ (0:\ ),\ 0.997 - 0.0743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.251620213 - 0.08379693525i\)
\(L(\frac12)\) \(\approx\) \(2.251620213 - 0.08379693525i\)
\(L(1)\) \(\approx\) \(1.409786691 + 0.02918398000i\)
\(L(1)\) \(\approx\) \(1.409786691 + 0.02918398000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.982 - 0.188i)T \)
7 \( 1 + (0.942 - 0.334i)T \)
11 \( 1 + (0.553 + 0.832i)T \)
13 \( 1 + (-0.800 + 0.599i)T \)
17 \( 1 + (-0.822 + 0.569i)T \)
19 \( 1 + (0.0189 - 0.999i)T \)
23 \( 1 + (-0.993 - 0.113i)T \)
29 \( 1 + (0.993 - 0.113i)T \)
31 \( 1 + (0.243 - 0.969i)T \)
37 \( 1 + (-0.521 - 0.853i)T \)
41 \( 1 + (0.881 + 0.472i)T \)
43 \( 1 + (-0.726 + 0.686i)T \)
47 \( 1 + (0.988 + 0.150i)T \)
53 \( 1 + (-0.206 - 0.978i)T \)
59 \( 1 + (0.822 + 0.569i)T \)
61 \( 1 + (0.997 + 0.0756i)T \)
67 \( 1 + (0.982 + 0.188i)T \)
71 \( 1 + (0.614 + 0.788i)T \)
73 \( 1 + (0.421 - 0.906i)T \)
79 \( 1 + (0.965 + 0.261i)T \)
83 \( 1 + (0.280 + 0.959i)T \)
89 \( 1 + (0.644 - 0.764i)T \)
97 \( 1 + (-0.243 - 0.969i)T \)
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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

−20.127698787017710445107623153304, −19.105657509364302634345080148973, −18.41716468004392720836101559787, −17.62927551694015144788560094506, −17.3464681290945756551628338662, −16.37316481485786088425706998942, −15.54864786195220849631842365515, −14.66414506106423781691278359777, −14.00532858890634115045743093246, −13.69977332119467143737750348405, −12.42002243822943212761833769321, −11.94875892357094279596519945809, −10.96131028140136325033384091757, −10.33137436224532009163689136704, −9.55007330927610043471442624391, −8.66726767392179276266945610828, −8.10729323936911394963162066802, −7.02222188646226877642358504211, −6.23446787173236898997858558427, −5.43782207975481263091545288468, −4.85378424485383505522568060026, −3.70886983624873539440589436377, −2.64621266168152452619213807613, −1.96925007186095255959013850101, −0.97705857667564555761716008076, 0.95564117035108086533549476291, 2.12920281541282069215079643033, 2.28733748379624011393824787002, 4.05322787154435593210331660773, 4.58303396868241520901536331046, 5.32286362808147161287741745532, 6.42606361643331993715345942589, 6.951434649502108117999512633178, 7.94508015821991944769935595881, 8.80035680897086022067534817763, 9.55395254335367510713402434222, 10.15699304585894143076283147231, 11.06747794150432978002388378560, 11.79642224130736602621158198370, 12.61552770635421261006486665156, 13.40687307361101028875362866082, 14.17949023924120799209201706440, 14.62598526822049448560868514336, 15.46079009137705997613917231370, 16.48135484002430663337557541198, 17.32592695402456498034586589454, 17.599608148297320164228218933320, 18.170585332037765577169864723472, 19.42969690943876611242793960587, 19.92836321620570773683689807853

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