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Properties

Label 1-6003-6003.1049-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.935 + 0.353i$
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  + (−0.984 − 0.175i)2-s + (0.938 + 0.346i)4-s + (−0.833 + 0.552i)5-s + (0.802 − 0.596i)7-s + (−0.862 − 0.505i)8-s + (0.917 − 0.396i)10-s + (0.912 + 0.409i)11-s + (0.464 + 0.885i)13-s + (−0.894 + 0.446i)14-s + (0.760 + 0.649i)16-s + (−0.415 − 0.909i)17-s + (−0.768 + 0.639i)19-s + (−0.973 + 0.229i)20-s + (−0.826 − 0.563i)22-s + (0.390 − 0.920i)25-s + (−0.301 − 0.953i)26-s + ⋯
L(s)  = 1  + (−0.984 − 0.175i)2-s + (0.938 + 0.346i)4-s + (−0.833 + 0.552i)5-s + (0.802 − 0.596i)7-s + (−0.862 − 0.505i)8-s + (0.917 − 0.396i)10-s + (0.912 + 0.409i)11-s + (0.464 + 0.885i)13-s + (−0.894 + 0.446i)14-s + (0.760 + 0.649i)16-s + (−0.415 − 0.909i)17-s + (−0.768 + 0.639i)19-s + (−0.973 + 0.229i)20-s + (−0.826 − 0.563i)22-s + (0.390 − 0.920i)25-s + (−0.301 − 0.953i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.935 + 0.353i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.935 + 0.353i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06634584186 + 0.3629988850i\)
\(L(\frac12)\) \(\approx\) \(0.06634584186 + 0.3629988850i\)
\(L(1)\) \(\approx\) \(0.6127546820 + 0.07102524315i\)
\(L(1)\) \(\approx\) \(0.6127546820 + 0.07102524315i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.984 - 0.175i)T \)
5 \( 1 + (-0.833 + 0.552i)T \)
7 \( 1 + (0.802 - 0.596i)T \)
11 \( 1 + (0.912 + 0.409i)T \)
13 \( 1 + (0.464 + 0.885i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (-0.768 + 0.639i)T \)
31 \( 1 + (0.427 + 0.903i)T \)
37 \( 1 + (-0.818 + 0.574i)T \)
41 \( 1 + (0.723 + 0.690i)T \)
43 \( 1 + (-0.427 + 0.903i)T \)
47 \( 1 + (-0.988 + 0.149i)T \)
53 \( 1 + (0.882 - 0.470i)T \)
59 \( 1 + (-0.981 - 0.189i)T \)
61 \( 1 + (-0.997 + 0.0679i)T \)
67 \( 1 + (0.912 - 0.409i)T \)
71 \( 1 + (-0.947 - 0.320i)T \)
73 \( 1 + (0.979 + 0.202i)T \)
79 \( 1 + (-0.942 - 0.333i)T \)
83 \( 1 + (-0.951 + 0.307i)T \)
89 \( 1 + (0.933 + 0.359i)T \)
97 \( 1 + (-0.634 + 0.773i)T \)
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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

−17.30243466678181216827344136620, −17.02058066938392660592336290551, −16.13679153318120031955797517751, −15.3687029182901421857027603774, −15.21259031970340262882158882342, −14.463233506438977511080443338510, −13.46296873477154539257999316792, −12.51498551920845881278670949580, −12.07466008483804799518410414559, −11.18478280774957864274451331768, −11.02493419852734424100504702477, −10.13243777944607680600768442965, −8.99021647580984952150146396812, −8.780571950589894321182814732866, −8.21249030215255249019426891224, −7.597799703304696455992159103575, −6.74632001064709636560521289017, −5.96112371796452075448471234796, −5.377505887846633139358363265013, −4.37866335157721795075998648439, −3.68364102844569629524790439254, −2.69406729867386696149707908440, −1.79360004888777963888962051874, −1.10688684373837587097164539252, −0.14339438677244385319739994649, 1.18460364279483691012247383809, 1.69011083302607466287998889175, 2.67479182584273120616834827452, 3.551219167427332503507325739564, 4.19559050053784943860743700258, 4.8527692509446810768615532336, 6.33788736924215840140200479885, 6.70942097693572609118577178042, 7.30844882917243711487950970376, 8.07445651518573760286924060276, 8.55496241893950729778076779025, 9.34177401424240748551496157121, 10.06979787064284265014706821074, 10.851690219783494592512448128688, 11.26353198977035674350330140803, 11.86269350193136087806027695115, 12.33085824932022514033864619467, 13.49304383684406972912075938058, 14.32760738267837144862266996472, 14.74877215317626743710093575470, 15.48961555129795265180265756345, 16.27143803786026941165095224940, 16.67447866064802695122419417644, 17.486030448497027298464652280327, 18.01070268974190476405310324897

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