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Properties

Label 1-6003-6003.1222-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.662 + 0.749i$
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  + (0.352 + 0.935i)2-s + (−0.751 + 0.659i)4-s + (−0.675 − 0.737i)5-s + (−0.546 + 0.837i)7-s + (−0.882 − 0.470i)8-s + (0.452 − 0.891i)10-s + (0.999 − 0.0135i)11-s + (0.314 − 0.949i)13-s + (−0.976 − 0.215i)14-s + (0.128 − 0.991i)16-s + (−0.841 − 0.540i)17-s + (−0.947 − 0.320i)19-s + (0.994 + 0.108i)20-s + (0.365 + 0.930i)22-s + (−0.0882 + 0.996i)25-s + (0.999 − 0.0407i)26-s + ⋯
L(s)  = 1  + (0.352 + 0.935i)2-s + (−0.751 + 0.659i)4-s + (−0.675 − 0.737i)5-s + (−0.546 + 0.837i)7-s + (−0.882 − 0.470i)8-s + (0.452 − 0.891i)10-s + (0.999 − 0.0135i)11-s + (0.314 − 0.949i)13-s + (−0.976 − 0.215i)14-s + (0.128 − 0.991i)16-s + (−0.841 − 0.540i)17-s + (−0.947 − 0.320i)19-s + (0.994 + 0.108i)20-s + (0.365 + 0.930i)22-s + (−0.0882 + 0.996i)25-s + (0.999 − 0.0407i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3044322448 + 0.6751314795i\)
\(L(\frac12)\) \(\approx\) \(0.3044322448 + 0.6751314795i\)
\(L(1)\) \(\approx\) \(0.7569123753 + 0.3408431448i\)
\(L(1)\) \(\approx\) \(0.7569123753 + 0.3408431448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.352 + 0.935i)T \)
5 \( 1 + (-0.675 - 0.737i)T \)
7 \( 1 + (-0.546 + 0.837i)T \)
11 \( 1 + (0.999 - 0.0135i)T \)
13 \( 1 + (0.314 - 0.949i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (-0.947 - 0.320i)T \)
31 \( 1 + (0.115 - 0.993i)T \)
37 \( 1 + (-0.996 + 0.0815i)T \)
41 \( 1 + (-0.928 - 0.371i)T \)
43 \( 1 + (0.115 + 0.993i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (-0.768 + 0.639i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (-0.760 + 0.649i)T \)
67 \( 1 + (-0.999 - 0.0135i)T \)
71 \( 1 + (-0.0611 - 0.998i)T \)
73 \( 1 + (0.523 - 0.852i)T \)
79 \( 1 + (0.923 + 0.384i)T \)
83 \( 1 + (0.869 + 0.494i)T \)
89 \( 1 + (-0.262 - 0.965i)T \)
97 \( 1 + (0.973 + 0.229i)T \)
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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

−17.49849578701246065940653253355, −16.96468445156207678931256306514, −16.115579176529087085773272736141, −15.330934090601479758855412989544, −14.69561926459522737002309134565, −14.042789924539680086235891546522, −13.64950348028396306981792966175, −12.74441455158999116237549261812, −12.1503561094277704032475540135, −11.49440238557090941724293413569, −10.86618244963840167610459399748, −10.43063936893665042760251499181, −9.69525570626802373949002585080, −8.849089747794195344661843535787, −8.363388665043265115635534794037, −7.17144708987122973562570257150, −6.50501893143723781294793702638, −6.25152340842364239460769817890, −4.798853835334523777610963355010, −4.255435703588263002619676610761, −3.60809452801323305002145851136, −3.26309041048908810621623345424, −2.03674521743937795635036969572, −1.51353365518906794885791053335, −0.2509958614349346483472445454, 0.64299186293833840832145242196, 1.93261611718415262924027476056, 3.1114166760011075256082872715, 3.57258082163336573251743298395, 4.5992098397435778364938680217, 4.86324250740338935918267821145, 6.00741818539568876810512902668, 6.267367315568785539610397939150, 7.16486228248350679631101410211, 7.93352831722862852936860981443, 8.59423838879806095354330947049, 9.07115420894369172259184364198, 9.57126355331143186089875542510, 10.75833795808879311790861912228, 11.69414383147085578456557555343, 12.1298870923125210026552670478, 12.87623484949701264886100535517, 13.239698242020886966681547011, 14.05982504134697769387811350087, 15.08486789633081747496845289217, 15.30162052319763876092757893756, 15.838868856423882862073955110466, 16.62964245825209755027074511785, 17.04145756113896679496898799844, 17.83585641658810784978043689468

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