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Properties

Label 1-6003-6003.1264-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.873 - 0.486i$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + i·8-s − i·10-s + (−0.866 − 0.5i)11-s + (0.5 + 0.866i)13-s + (0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s + i·17-s + i·19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)25-s + i·26-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + i·8-s − i·10-s + (−0.866 − 0.5i)11-s + (0.5 + 0.866i)13-s + (0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s + i·17-s + i·19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)25-s + i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.873 - 0.486i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.873 - 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005647896336 + 0.02176098914i\)
\(L(\frac12)\) \(\approx\) \(0.005647896336 + 0.02176098914i\)
\(L(1)\) \(\approx\) \(1.270197476 + 0.2580040820i\)
\(L(1)\) \(\approx\) \(1.270197476 + 0.2580040820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + iT \)
19 \( 1 + iT \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + iT \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−17.709468922963212269752070132238, −16.28491875912720341969063830614, −15.75221990353085521756197367346, −15.12325952381396269384698036096, −14.92097981430897112117182094428, −14.05517165044411059537051817883, −13.30248889519690361605847111820, −12.78803050806615578425723540264, −11.9912819449785210570658264393, −11.32908830269673324866102526009, −11.04423821058801809745939463843, −10.21490362999387883380690155329, −9.58274601176860236923121817727, −8.65066833769676230380459770893, −7.68945279877739450889349209705, −7.28757309286828255435861160714, −6.28509692386558897694172264344, −5.75275890887974091933300984531, −4.83921206167824058422119676695, −4.53522617551031849628956914004, −3.2373178154650664995430034475, −2.91088817438732793435326299026, −2.30109424163273099450016038089, −1.31093711660046193854130497013, −0.00368068575855673523480326294, 1.450227228783056720722867533318, 1.968223178796277155795712401327, 3.41592024085383989714244196377, 3.75227440006972299335560647257, 4.47728913878436069111069370998, 5.1078633958780939409092456955, 5.81202953701916364688359035033, 6.53213629081803218743132687073, 7.493963804675991775298902994845, 7.89932027203251464935947743082, 8.49408271963497456788519968976, 9.182593038856297402154945799161, 10.45605557889303632413252733141, 10.92845174254217885206100379376, 11.62969719451496018861892161541, 12.339436093377440242974248563400, 12.99188505516572792868578630198, 13.425887734090274859675379735703, 14.27814934604698268457811781312, 14.61111228072643016879316644836, 15.65875000406041803468925546741, 16.06974473966924447035439111153, 16.67652194868126815401330266275, 17.08775321861322760239222078198, 17.902831783646663725218642840149

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