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Properties

Label 1-6003-6003.430-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.291 - 0.956i$
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  + (−0.427 − 0.903i)2-s + (−0.634 + 0.773i)4-s + (0.994 − 0.108i)5-s + (0.999 + 0.0271i)7-s + (0.970 + 0.242i)8-s + (−0.523 − 0.852i)10-s + (0.869 + 0.494i)11-s + (−0.912 − 0.409i)13-s + (−0.403 − 0.915i)14-s + (−0.195 − 0.980i)16-s + (−0.959 − 0.281i)17-s + (0.986 + 0.162i)19-s + (−0.546 + 0.837i)20-s + (0.0747 − 0.997i)22-s + (0.976 − 0.215i)25-s + (0.0203 + 0.999i)26-s + ⋯
L(s)  = 1  + (−0.427 − 0.903i)2-s + (−0.634 + 0.773i)4-s + (0.994 − 0.108i)5-s + (0.999 + 0.0271i)7-s + (0.970 + 0.242i)8-s + (−0.523 − 0.852i)10-s + (0.869 + 0.494i)11-s + (−0.912 − 0.409i)13-s + (−0.403 − 0.915i)14-s + (−0.195 − 0.980i)16-s + (−0.959 − 0.281i)17-s + (0.986 + 0.162i)19-s + (−0.546 + 0.837i)20-s + (0.0747 − 0.997i)22-s + (0.976 − 0.215i)25-s + (0.0203 + 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.291 - 0.956i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (430, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.291 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.643704146 - 1.217884236i\)
\(L(\frac12)\) \(\approx\) \(1.643704146 - 1.217884236i\)
\(L(1)\) \(\approx\) \(1.059270803 - 0.4587194010i\)
\(L(1)\) \(\approx\) \(1.059270803 - 0.4587194010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.427 - 0.903i)T \)
5 \( 1 + (0.994 - 0.108i)T \)
7 \( 1 + (0.999 + 0.0271i)T \)
11 \( 1 + (0.869 + 0.494i)T \)
13 \( 1 + (-0.912 - 0.409i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (0.986 + 0.162i)T \)
31 \( 1 + (0.314 - 0.949i)T \)
37 \( 1 + (-0.999 + 0.0407i)T \)
41 \( 1 + (-0.327 - 0.945i)T \)
43 \( 1 + (0.314 + 0.949i)T \)
47 \( 1 + (0.365 - 0.930i)T \)
53 \( 1 + (0.339 + 0.940i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (-0.169 + 0.985i)T \)
67 \( 1 + (0.869 - 0.494i)T \)
71 \( 1 + (0.685 - 0.728i)T \)
73 \( 1 + (0.488 + 0.872i)T \)
79 \( 1 + (-0.751 - 0.659i)T \)
83 \( 1 + (0.704 - 0.709i)T \)
89 \( 1 + (0.794 + 0.607i)T \)
97 \( 1 + (-0.802 - 0.596i)T \)
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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

−17.591150813316572710278928882, −17.2952550564466070054850354735, −16.77711686940533770780527770655, −15.905299792485550346392688497808, −15.24122135973702047541168222238, −14.41382156394618059366196986551, −14.121336560416588662257412859996, −13.67760387218413115788472177007, −12.73761066349091460672177088731, −11.81974002203678811991576590002, −11.07608622221458201751521485037, −10.44693039046795880032758931274, −9.65092259278536405690094877512, −9.11802524644732691641465013039, −8.56680204710490052034764336046, −7.792428507620944513338566122, −6.87975017898613932385792225235, −6.606978291989249161688306102315, −5.66048643036455514100436601882, −5.04492119186983277866722517293, −4.549185323655529635312823131125, −3.50171273921624112217056002860, −2.27239754618083153893623509209, −1.63859226284000704029352505567, −0.89203871620496229437985404222, 0.72236950729394773611868595467, 1.56739211395486261513377298952, 2.136514831322585328770486362606, 2.74982758313320940492372676728, 3.80440170925508199232286027048, 4.653878873367616288646698188326, 5.06461833206106618574601169706, 5.95250386405146841867164815960, 7.07206457703702623771067901241, 7.52355977029500283161961870393, 8.48036730355134867485800175805, 9.09926161270706315983877799218, 9.60760975528129348039788458402, 10.27448044513464293943203543644, 10.89099159768404734704075108958, 11.73267900389005712373962559130, 12.10816199519180667531046164850, 12.86837868740365236361358580097, 13.75012365769366445953340338495, 14.00587940451679624899888321898, 14.82343904038075084106660688418, 15.566049014603849110943436012797, 16.7839109828091620556241034609, 17.092338964799304310770289157232, 17.72216890156448700116453858700

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