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Properties

Label 1-6003-6003.20-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.739 - 0.673i$
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  + (−0.990 + 0.135i)2-s + (0.963 − 0.268i)4-s + (0.869 − 0.494i)5-s + (−0.128 + 0.991i)7-s + (−0.917 + 0.396i)8-s + (−0.794 + 0.607i)10-s + (−0.634 − 0.773i)11-s + (0.938 − 0.346i)13-s + (−0.00679 − 0.999i)14-s + (0.855 − 0.517i)16-s + (0.841 + 0.540i)17-s + (0.714 + 0.699i)19-s + (0.704 − 0.709i)20-s + (0.733 + 0.680i)22-s + (0.511 − 0.859i)25-s + (−0.882 + 0.470i)26-s + ⋯
L(s)  = 1  + (−0.990 + 0.135i)2-s + (0.963 − 0.268i)4-s + (0.869 − 0.494i)5-s + (−0.128 + 0.991i)7-s + (−0.917 + 0.396i)8-s + (−0.794 + 0.607i)10-s + (−0.634 − 0.773i)11-s + (0.938 − 0.346i)13-s + (−0.00679 − 0.999i)14-s + (0.855 − 0.517i)16-s + (0.841 + 0.540i)17-s + (0.714 + 0.699i)19-s + (0.704 − 0.709i)20-s + (0.733 + 0.680i)22-s + (0.511 − 0.859i)25-s + (−0.882 + 0.470i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.739 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243510452 - 0.4817196994i\)
\(L(\frac12)\) \(\approx\) \(1.243510452 - 0.4817196994i\)
\(L(1)\) \(\approx\) \(0.8511207306 - 0.03779147923i\)
\(L(1)\) \(\approx\) \(0.8511207306 - 0.03779147923i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.990 + 0.135i)T \)
5 \( 1 + (0.869 - 0.494i)T \)
7 \( 1 + (-0.128 + 0.991i)T \)
11 \( 1 + (-0.634 - 0.773i)T \)
13 \( 1 + (0.938 - 0.346i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (0.714 + 0.699i)T \)
31 \( 1 + (-0.942 - 0.333i)T \)
37 \( 1 + (-0.557 + 0.830i)T \)
41 \( 1 + (-0.928 - 0.371i)T \)
43 \( 1 + (0.942 - 0.333i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.452 - 0.891i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (0.403 - 0.915i)T \)
67 \( 1 + (0.634 - 0.773i)T \)
71 \( 1 + (-0.742 + 0.670i)T \)
73 \( 1 + (0.947 + 0.320i)T \)
79 \( 1 + (0.876 - 0.482i)T \)
83 \( 1 + (-0.568 - 0.822i)T \)
89 \( 1 + (-0.999 + 0.0407i)T \)
97 \( 1 + (-0.777 - 0.628i)T \)
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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

−17.96727539867829870378789183680, −17.31242319272676252892379129211, −16.51465626425243911812828178035, −16.1071044006435127910471411994, −15.34129007592832055675325837149, −14.47158705720911702807942662722, −13.90475789683659587852876569141, −13.16491523440696561240544235168, −12.558919543897700853955856082387, −11.57290422221392063756295831245, −10.90322373822794915308451026465, −10.50307372751106512906722064948, −9.68638776724205514408557607063, −9.456437868788266729047957050212, −8.50497453745590268188902498721, −7.57174561050959000014197340853, −7.17325231397421322226087977139, −6.5952547185882736775557021284, −5.73345438171925665752770627905, −4.99912132856157462265010251484, −3.81308244241972263793101983760, −3.13950077263854826947573521743, −2.39558835145223159032633922222, −1.52905577418333200055590341038, −0.91042898229138459653099895552, 0.5352952801093034477250321476, 1.529346278277415221330254572316, 2.0044223818505357452398856671, 3.04653631547385399273781689354, 3.53224953213767601572800041144, 5.19410825032372325782603833397, 5.5942413980946539753560877047, 6.02029386179969926113912822847, 6.81711229038884739232175731253, 7.913934397104208529675378084691, 8.43358801760093821439296233218, 8.793389353331580235080954227070, 9.78204051202679512830825215356, 10.028412206945190276993645524711, 10.92730960849231964574236780262, 11.53829699050720485040196965721, 12.40527105261046287101366872331, 12.86751703651296949141711624663, 13.7690503149354698602137630382, 14.425508240262631802103880031583, 15.27865375829754084670458712853, 15.85411710827615665816220216549, 16.46252788336733606217116870772, 16.880545671890641122033043470342, 17.86994049340408312748516769319

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