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Properties

Label 1-6003-6003.2230-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.116 - 0.993i$
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  + (−0.563 + 0.826i)2-s + (−0.365 − 0.930i)4-s + (0.0747 − 0.997i)5-s + (−0.365 + 0.930i)7-s + (0.974 + 0.222i)8-s + (0.781 + 0.623i)10-s + (0.680 + 0.733i)11-s + (−0.955 − 0.294i)13-s + (−0.563 − 0.826i)14-s + (−0.733 + 0.680i)16-s − i·17-s + (−0.781 − 0.623i)19-s + (−0.955 + 0.294i)20-s + (−0.988 + 0.149i)22-s + (−0.988 − 0.149i)25-s + (0.781 − 0.623i)26-s + ⋯
L(s)  = 1  + (−0.563 + 0.826i)2-s + (−0.365 − 0.930i)4-s + (0.0747 − 0.997i)5-s + (−0.365 + 0.930i)7-s + (0.974 + 0.222i)8-s + (0.781 + 0.623i)10-s + (0.680 + 0.733i)11-s + (−0.955 − 0.294i)13-s + (−0.563 − 0.826i)14-s + (−0.733 + 0.680i)16-s − i·17-s + (−0.781 − 0.623i)19-s + (−0.955 + 0.294i)20-s + (−0.988 + 0.149i)22-s + (−0.988 − 0.149i)25-s + (0.781 − 0.623i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.116 - 0.993i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (2230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.116 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3246446699 - 0.3648993040i\)
\(L(\frac12)\) \(\approx\) \(0.3246446699 - 0.3648993040i\)
\(L(1)\) \(\approx\) \(0.6624347989 + 0.1103006866i\)
\(L(1)\) \(\approx\) \(0.6624347989 + 0.1103006866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.563 + 0.826i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
7 \( 1 + (-0.365 + 0.930i)T \)
11 \( 1 + (0.680 + 0.733i)T \)
13 \( 1 + (-0.955 - 0.294i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.781 - 0.623i)T \)
31 \( 1 + (0.997 + 0.0747i)T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (0.997 - 0.0747i)T \)
47 \( 1 + (0.680 + 0.733i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.930 - 0.365i)T \)
67 \( 1 + (-0.733 - 0.680i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.433 + 0.900i)T \)
79 \( 1 + (-0.294 - 0.955i)T \)
83 \( 1 + (0.988 + 0.149i)T \)
89 \( 1 + (0.433 + 0.900i)T \)
97 \( 1 + (0.149 - 0.988i)T \)
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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

−17.828381574894324831035114124877, −17.23358691899494848072364321416, −16.923657145896497740276778307, −16.18233650145343011517760370915, −15.203093640236694006172043524272, −14.43068553004166667651787871667, −13.88663091560269548018028247004, −13.34797019961789410502004699471, −12.35739051171202565666852900133, −11.98030816305335948677700655309, −11.02456101020651144454039651597, −10.55728992693620667209447950732, −10.17354051530698990091837360848, −9.387887819259499367550217390650, −8.67268900964329860828740655064, −7.822999101046129632736536405382, −7.2725651639225283837609551186, −6.54183011599578835658849826272, −5.93066462230810607381199850249, −4.56181635146921939349196930839, −3.90913577223994746859867418426, −3.44438508178144578141545927639, −2.58287066026835081237411030573, −1.87407484546174958053758702254, −0.94229159752976275875596539508, 0.181861270994664805080171789568, 1.14219218406837971482963310938, 2.09795109053144802868718522637, 2.756913166527284135874681196214, 4.26431552255135179947686760925, 4.68501723637569069563722089472, 5.39939780271543034461506024861, 6.02536664673159913690825964249, 6.835230628480370335983995558936, 7.44620606783679001280963869602, 8.24660670492170712764311092172, 8.99281904912559131388040229555, 9.33161785551591288934369170613, 9.85472700847548981067434568626, 10.73360278261132059913672071028, 11.80498847951401204824153785207, 12.26637587022266277665041510660, 12.9250427088296537458819439950, 13.7207330011386032225936956033, 14.43179381753412148526159702121, 15.18956505159243537685800227672, 15.58271412765741458237231782196, 16.2691255073227957059100798674, 16.88329877577712508032007126497, 17.63307297547772052793468987539

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