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Properties

Label 1-2004-2004.1727-r0-0-0
Degree $1$
Conductor $2004$
Sign $-0.151 + 0.988i$
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  + (−0.862 + 0.505i)5-s + (−0.954 − 0.298i)7-s + (−0.644 − 0.764i)11-s + (−0.387 + 0.922i)13-s + (0.982 + 0.188i)17-s + (−0.776 − 0.629i)19-s + (−0.584 − 0.811i)23-s + (0.489 − 0.872i)25-s + (0.584 − 0.811i)29-s + (0.843 − 0.537i)31-s + (0.974 − 0.225i)35-s + (0.614 − 0.788i)37-s + (−0.421 + 0.906i)41-s + (0.521 + 0.853i)43-s + (0.672 − 0.739i)47-s + ⋯
L(s)  = 1  + (−0.862 + 0.505i)5-s + (−0.954 − 0.298i)7-s + (−0.644 − 0.764i)11-s + (−0.387 + 0.922i)13-s + (0.982 + 0.188i)17-s + (−0.776 − 0.629i)19-s + (−0.584 − 0.811i)23-s + (0.489 − 0.872i)25-s + (0.584 − 0.811i)29-s + (0.843 − 0.537i)31-s + (0.974 − 0.225i)35-s + (0.614 − 0.788i)37-s + (−0.421 + 0.906i)41-s + (0.521 + 0.853i)43-s + (0.672 − 0.739i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
Sign: $-0.151 + 0.988i$
Analytic conductor: \(9.30653\)
Root analytic conductor: \(9.30653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2004} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2004,\ (0:\ ),\ -0.151 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3267437324 + 0.3804975807i\)
\(L(\frac12)\) \(\approx\) \(0.3267437324 + 0.3804975807i\)
\(L(1)\) \(\approx\) \(0.6884880916 + 0.03906722500i\)
\(L(1)\) \(\approx\) \(0.6884880916 + 0.03906722500i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.862 + 0.505i)T \)
7 \( 1 + (-0.954 - 0.298i)T \)
11 \( 1 + (-0.644 - 0.764i)T \)
13 \( 1 + (-0.387 + 0.922i)T \)
17 \( 1 + (0.982 + 0.188i)T \)
19 \( 1 + (-0.776 - 0.629i)T \)
23 \( 1 + (-0.584 - 0.811i)T \)
29 \( 1 + (0.584 - 0.811i)T \)
31 \( 1 + (0.843 - 0.537i)T \)
37 \( 1 + (0.614 - 0.788i)T \)
41 \( 1 + (-0.421 + 0.906i)T \)
43 \( 1 + (0.521 + 0.853i)T \)
47 \( 1 + (0.672 - 0.739i)T \)
53 \( 1 + (-0.351 + 0.936i)T \)
59 \( 1 + (-0.982 + 0.188i)T \)
61 \( 1 + (-0.914 + 0.404i)T \)
67 \( 1 + (-0.862 - 0.505i)T \)
71 \( 1 + (0.280 + 0.959i)T \)
73 \( 1 + (-0.999 + 0.0378i)T \)
79 \( 1 + (0.993 + 0.113i)T \)
83 \( 1 + (-0.700 + 0.713i)T \)
89 \( 1 + (-0.997 - 0.0756i)T \)
97 \( 1 + (-0.843 - 0.537i)T \)
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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

−19.69393448414477439436401529015, −19.13736990513792943318233592163, −18.42933591809070947945902975822, −17.50751565875852010859015954138, −16.76561592659399889027493344202, −15.96032048312484629916772451186, −15.50041286438653360650444442234, −14.86849978119069681537966085034, −13.80665721072202634776315378450, −12.881409433405455970825323769100, −12.2959464764542382678905547418, −12.03692043889227690544517936998, −10.65579146396434557994170937309, −10.11742707623880047401133981846, −9.34383775316664308884610594929, −8.36462159421992126276410593535, −7.75354315058569230175856774837, −7.0524543564252377422198756218, −5.96203990849549692186938839630, −5.22154291315294932656309652901, −4.39259570428834520746947845782, −3.39294476719257024642308567803, −2.80916068996116230092020855290, −1.51837216160532022465994627488, −0.23272662896272588243227263569, 0.82180099239813878309321964933, 2.502121121862573142602296801217, 2.98350818375688185011816639410, 4.046891653820284876944022674631, 4.5485123285192004438518803107, 6.00311173380760469543987734425, 6.43503622786767205290445146727, 7.402920923893267538753295612147, 8.01867628404314249810363161454, 8.88623849578876349997181438312, 9.870794861990484433217029711311, 10.49183833202801203820565783923, 11.25420417383437533876420154636, 12.05648544747797370021916715849, 12.697756312560963614144130907448, 13.631859691297584468807324849730, 14.24491414569833399624494859267, 15.1546722007025972133210183547, 15.78183423845619533287440974503, 16.54153113520432762428299010850, 16.940360426253567044053621550339, 18.25252891053616626784789739519, 18.80962933218683305212801779130, 19.4135557264418568426022926021, 19.831622169962704134675884223410

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