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Properties

Label 1-6003-6003.1003-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.203 + 0.979i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (0.998 − 0.0475i)2-s + (0.995 − 0.0950i)4-s + (0.723 + 0.690i)5-s + (−0.981 + 0.189i)7-s + (0.989 − 0.142i)8-s + (0.755 + 0.654i)10-s + (−0.458 − 0.888i)11-s + (−0.981 − 0.189i)13-s + (−0.971 + 0.235i)14-s + (0.981 − 0.189i)16-s + (−0.909 + 0.415i)17-s + (0.909 + 0.415i)19-s + (0.786 + 0.618i)20-s + (−0.5 − 0.866i)22-s + (0.0475 + 0.998i)25-s + (−0.989 − 0.142i)26-s + ⋯
L(s)  = 1  + (0.998 − 0.0475i)2-s + (0.995 − 0.0950i)4-s + (0.723 + 0.690i)5-s + (−0.981 + 0.189i)7-s + (0.989 − 0.142i)8-s + (0.755 + 0.654i)10-s + (−0.458 − 0.888i)11-s + (−0.981 − 0.189i)13-s + (−0.971 + 0.235i)14-s + (0.981 − 0.189i)16-s + (−0.909 + 0.415i)17-s + (0.909 + 0.415i)19-s + (0.786 + 0.618i)20-s + (−0.5 − 0.866i)22-s + (0.0475 + 0.998i)25-s + (−0.989 − 0.142i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.203 + 0.979i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.203 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.659930252 + 2.040285112i\)
\(L(\frac12)\) \(\approx\) \(1.659930252 + 2.040285112i\)
\(L(1)\) \(\approx\) \(1.739767883 + 0.3530563131i\)
\(L(1)\) \(\approx\) \(1.739767883 + 0.3530563131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.998 - 0.0475i)T \)
5 \( 1 + (0.723 + 0.690i)T \)
7 \( 1 + (-0.981 + 0.189i)T \)
11 \( 1 + (-0.458 - 0.888i)T \)
13 \( 1 + (-0.981 - 0.189i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.909 + 0.415i)T \)
31 \( 1 + (0.618 + 0.786i)T \)
37 \( 1 + (-0.281 + 0.959i)T \)
41 \( 1 + (-0.690 + 0.723i)T \)
43 \( 1 + (0.618 - 0.786i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.981 + 0.189i)T \)
61 \( 1 + (-0.371 + 0.928i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (0.945 - 0.327i)T \)
83 \( 1 + (-0.723 + 0.690i)T \)
89 \( 1 + (-0.989 - 0.142i)T \)
97 \( 1 + (-0.971 - 0.235i)T \)
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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

−17.585053807703732941754274255, −16.5824980725062258369222394240, −16.24301578288389937162837508831, −15.50600537694678756847489600862, −14.96106614501552579674493750721, −13.98186809422915651020569986425, −13.60625052684860826748732959176, −12.9040304300919935106996723053, −12.52095085160598021024636067219, −11.86479787350267005292620263571, −11.036146226787492526734258734248, −10.0715004953969567210224249548, −9.73135616123034428465839985100, −9.04761379730773067983283311476, −7.93008108743976109357365424500, −7.146600935376618374281601444948, −6.72751539502510794610188638612, −5.87844165894060354554846936874, −5.17812730420809793133398329916, −4.65572236044637117045425829962, −3.98347802723172086522954869165, −2.8787853407885364863943088197, −2.41212608620889193882786594539, −1.66090849328044021366693629161, −0.40198508923468491061396767081, 1.1855754818508603393926513443, 2.219056660722328824088684460710, 2.872570060734047990497567339920, 3.22875067882549326888197726558, 4.14273140469713317265650502405, 5.2218109656014293211777670637, 5.58314321208910990423034971329, 6.43104221882426700068596100316, 6.79527316552810604781144237899, 7.57337956649540850370979875431, 8.515448560060110312133037527007, 9.45664183103430310197347618471, 10.3109651986445471449673587576, 10.40736355818351139688233125043, 11.4723113856402168459859150402, 12.02367873993846817879212201168, 12.82197623926715659337858017695, 13.45151566471572339249748799707, 13.778730417910881755857313219459, 14.54952498832297942526525686949, 15.27572266878090370561137765704, 15.70022688851894182143433741507, 16.55382865792272880896913762958, 17.00145063935874594291166297535, 17.96824088050805497857716999573

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