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Properties

Label 1-6003-6003.4246-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.724 + 0.688i$
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  + (0.458 − 0.888i)2-s + (−0.580 − 0.814i)4-s + (0.235 − 0.971i)5-s + (0.327 − 0.945i)7-s + (−0.989 + 0.142i)8-s + (−0.755 − 0.654i)10-s + (−0.998 − 0.0475i)11-s + (0.327 + 0.945i)13-s + (−0.690 − 0.723i)14-s + (−0.327 + 0.945i)16-s + (0.909 − 0.415i)17-s + (−0.909 − 0.415i)19-s + (−0.928 + 0.371i)20-s + (−0.5 + 0.866i)22-s + (−0.888 − 0.458i)25-s + (0.989 + 0.142i)26-s + ⋯
L(s)  = 1  + (0.458 − 0.888i)2-s + (−0.580 − 0.814i)4-s + (0.235 − 0.971i)5-s + (0.327 − 0.945i)7-s + (−0.989 + 0.142i)8-s + (−0.755 − 0.654i)10-s + (−0.998 − 0.0475i)11-s + (0.327 + 0.945i)13-s + (−0.690 − 0.723i)14-s + (−0.327 + 0.945i)16-s + (0.909 − 0.415i)17-s + (−0.909 − 0.415i)19-s + (−0.928 + 0.371i)20-s + (−0.5 + 0.866i)22-s + (−0.888 − 0.458i)25-s + (0.989 + 0.142i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.724 + 0.688i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (4246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.724 + 0.688i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1491600719 - 0.05956821677i\)
\(L(\frac12)\) \(\approx\) \(-0.1491600719 - 0.05956821677i\)
\(L(1)\) \(\approx\) \(0.6978362019 - 0.7509049084i\)
\(L(1)\) \(\approx\) \(0.6978362019 - 0.7509049084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.458 - 0.888i)T \)
5 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (0.327 - 0.945i)T \)
11 \( 1 + (-0.998 - 0.0475i)T \)
13 \( 1 + (0.327 + 0.945i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.909 - 0.415i)T \)
31 \( 1 + (-0.371 + 0.928i)T \)
37 \( 1 + (0.281 - 0.959i)T \)
41 \( 1 + (-0.971 - 0.235i)T \)
43 \( 1 + (-0.371 - 0.928i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.327 - 0.945i)T \)
61 \( 1 + (0.618 + 0.786i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (0.189 - 0.981i)T \)
83 \( 1 + (-0.235 - 0.971i)T \)
89 \( 1 + (0.989 + 0.142i)T \)
97 \( 1 + (-0.690 + 0.723i)T \)
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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

−18.261006449828790155255248661757, −17.71500757082591246109730358219, −16.897193085712829341230122287693, −16.261418021344000692232080672762, −15.312803766303099212318178478800, −15.07022690350382416801389289435, −14.68758637571986466789618025039, −13.763850030020078295329483417510, −13.09933261873658812107302581797, −12.62928620304107024732223948702, −11.780324485311115544238528204176, −11.10521469187712307988577168334, −10.19250346471075234615882594037, −9.76238089847240289517130941904, −8.61911680447727632614701300350, −8.058221295425935236571304394557, −7.70710822368554996042962846642, −6.66599123689278199959436171025, −6.07617023701682058155857899885, −5.529704249857278491544633961998, −4.974728939512232640115835076215, −3.90257715736074268533703528245, −3.12104550291378503813443302366, −2.64730316560971412549689452066, −1.684206414179980873562293376795, 0.03465057221731664838555221879, 1.01399785075900441192470133043, 1.68177536719806374073971482167, 2.4185258067770416208618318090, 3.45192467331386089610571116340, 4.10881419480291810173373784458, 4.80197914803092848802200876545, 5.24874592748647528130279193486, 6.069808323897720018720316152769, 6.99911287881882179161794638833, 7.84312515841423891338837450009, 8.71053431630138433290490118497, 9.11565744090394675412268882662, 10.19412209946328032513991017235, 10.358872811319884410020551872347, 11.289196051672897797884010696517, 11.8537374960041087605874652046, 12.630936217293825203643179889754, 13.18572165912043728655343829684, 13.67488714967010268508613885429, 14.28204314668538081722473498519, 14.9487961132455140219261797193, 16.02836282103889420141770322604, 16.36165615274909974469565472761, 17.29481292288373223684561386770

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