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Properties

Label 1-6003-6003.1114-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.695 - 0.718i$
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  + (0.371 + 0.928i)2-s + (−0.723 + 0.690i)4-s + (0.981 − 0.189i)5-s + (−0.0475 + 0.998i)7-s + (−0.909 − 0.415i)8-s + (0.540 + 0.841i)10-s + (0.618 + 0.786i)11-s + (−0.0475 − 0.998i)13-s + (−0.945 + 0.327i)14-s + (0.0475 − 0.998i)16-s + (0.281 + 0.959i)17-s + (−0.281 + 0.959i)19-s + (−0.580 + 0.814i)20-s + (−0.5 + 0.866i)22-s + (0.928 − 0.371i)25-s + (0.909 − 0.415i)26-s + ⋯
L(s)  = 1  + (0.371 + 0.928i)2-s + (−0.723 + 0.690i)4-s + (0.981 − 0.189i)5-s + (−0.0475 + 0.998i)7-s + (−0.909 − 0.415i)8-s + (0.540 + 0.841i)10-s + (0.618 + 0.786i)11-s + (−0.0475 − 0.998i)13-s + (−0.945 + 0.327i)14-s + (0.0475 − 0.998i)16-s + (0.281 + 0.959i)17-s + (−0.281 + 0.959i)19-s + (−0.580 + 0.814i)20-s + (−0.5 + 0.866i)22-s + (0.928 − 0.371i)25-s + (0.909 − 0.415i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.695 - 0.718i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5138087416 + 1.211355068i\)
\(L(\frac12)\) \(\approx\) \(-0.5138087416 + 1.211355068i\)
\(L(1)\) \(\approx\) \(0.8829361264 + 0.8301866287i\)
\(L(1)\) \(\approx\) \(0.8829361264 + 0.8301866287i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.371 + 0.928i)T \)
5 \( 1 + (0.981 - 0.189i)T \)
7 \( 1 + (-0.0475 + 0.998i)T \)
11 \( 1 + (0.618 + 0.786i)T \)
13 \( 1 + (-0.0475 - 0.998i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (-0.281 + 0.959i)T \)
31 \( 1 + (-0.814 + 0.580i)T \)
37 \( 1 + (-0.755 - 0.654i)T \)
41 \( 1 + (-0.189 - 0.981i)T \)
43 \( 1 + (-0.814 - 0.580i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (0.0475 + 0.998i)T \)
61 \( 1 + (0.0950 + 0.995i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.281 + 0.959i)T \)
79 \( 1 + (-0.458 + 0.888i)T \)
83 \( 1 + (-0.981 - 0.189i)T \)
89 \( 1 + (0.909 - 0.415i)T \)
97 \( 1 + (-0.945 - 0.327i)T \)
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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

−17.386794597577940884448135794055, −16.75552341645504708119646119760, −16.19469867406376269706118503741, −15.00301401735528625413268861303, −14.32051191416553469533015685963, −13.95287118607142578729513845179, −13.36348738405530535371020946410, −12.95049654052940085352713009298, −11.82449094878613493619471516846, −11.32563447349044431854370203985, −10.82959081263434864383181126801, −9.98035027730041973882601534621, −9.482719069740853925687179434553, −8.99564699714011318722868205929, −8.06144587264624360044204255264, −6.752063606962218954522075826586, −6.6408596765834740078195102453, −5.61808646588176890829470498402, −4.85054347386465224042229365311, −4.29054843387534461076873259847, −3.33023109789877807062851468695, −2.87488899447651486740408391723, −1.780737086018528840317638400092, −1.331356344937552864710163810913, −0.263831179673795120919852250406, 1.42367058169264973783498587590, 2.08797183109346693142771397988, 3.0984673636525571149350062844, 3.79356243195288836158489432028, 4.74585009072118468045664995852, 5.520528977001421165905290376150, 5.77938209687851803455558644941, 6.532185645914421742655482490301, 7.23909439553672823591620942341, 8.1892338924362917994421243954, 8.66680844131867553245392871281, 9.33923606995355712771319415740, 10.001785871910180128561415157382, 10.64432039591700946682082404662, 11.9517752941195178732039634060, 12.49432985213541343587237460813, 12.7861537073434763943029159012, 13.59703683516285302127231379772, 14.50873978834673927028365964146, 14.73266204350298330936511357221, 15.393605205635525762954778421279, 16.16995427569857307767598939973, 16.824277507065911826233807256198, 17.493248650876044487127825487547, 17.838921259161574497009009723587

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