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Properties

Label 1-6003-6003.1118-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.230 - 0.973i$
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  + (0.195 + 0.980i)2-s + (−0.923 + 0.384i)4-s + (0.906 − 0.421i)5-s + (−0.994 − 0.108i)7-s + (−0.557 − 0.830i)8-s + (0.591 + 0.806i)10-s + (−0.476 + 0.879i)11-s + (−0.115 + 0.993i)13-s + (−0.0882 − 0.996i)14-s + (0.704 − 0.709i)16-s + (0.415 + 0.909i)17-s + (−0.794 − 0.607i)19-s + (−0.675 + 0.737i)20-s + (−0.955 − 0.294i)22-s + (0.644 − 0.764i)25-s + (−0.996 + 0.0815i)26-s + ⋯
L(s)  = 1  + (0.195 + 0.980i)2-s + (−0.923 + 0.384i)4-s + (0.906 − 0.421i)5-s + (−0.994 − 0.108i)7-s + (−0.557 − 0.830i)8-s + (0.591 + 0.806i)10-s + (−0.476 + 0.879i)11-s + (−0.115 + 0.993i)13-s + (−0.0882 − 0.996i)14-s + (0.704 − 0.709i)16-s + (0.415 + 0.909i)17-s + (−0.794 − 0.607i)19-s + (−0.675 + 0.737i)20-s + (−0.955 − 0.294i)22-s + (0.644 − 0.764i)25-s + (−0.996 + 0.0815i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.230 - 0.973i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2110157392 + 0.2667878317i\)
\(L(\frac12)\) \(\approx\) \(-0.2110157392 + 0.2667878317i\)
\(L(1)\) \(\approx\) \(0.7332740014 + 0.5087083151i\)
\(L(1)\) \(\approx\) \(0.7332740014 + 0.5087083151i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.195 + 0.980i)T \)
5 \( 1 + (0.906 - 0.421i)T \)
7 \( 1 + (-0.994 - 0.108i)T \)
11 \( 1 + (-0.476 + 0.879i)T \)
13 \( 1 + (-0.115 + 0.993i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (-0.794 - 0.607i)T \)
31 \( 1 + (0.288 + 0.957i)T \)
37 \( 1 + (-0.986 + 0.162i)T \)
41 \( 1 + (-0.235 + 0.971i)T \)
43 \( 1 + (-0.288 + 0.957i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (0.182 - 0.983i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (-0.777 - 0.628i)T \)
67 \( 1 + (0.476 + 0.879i)T \)
71 \( 1 + (0.992 - 0.122i)T \)
73 \( 1 + (-0.452 - 0.891i)T \)
79 \( 1 + (0.966 - 0.255i)T \)
83 \( 1 + (-0.999 + 0.0135i)T \)
89 \( 1 + (-0.862 + 0.505i)T \)
97 \( 1 + (0.833 - 0.552i)T \)
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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

−17.2442281659710643760004690535, −16.88354973481154180116039214124, −15.81615918865477789815069222127, −15.24422482613426087048509290327, −14.338774064996852012406583231554, −13.75742736276235239560109370836, −13.33881854552193808949498981476, −12.59993560436290273096706368303, −12.170196058907365948246662718141, −11.09355514745238088007887421377, −10.597362857542315530356016904660, −10.04420263349506682379883270039, −9.51660388656007014214679955895, −8.79203414236029841509183509555, −8.04611883648405447552736046009, −7.04905004986942501424952852799, −6.10562730410247621372704477679, −5.66049540197298934817579308610, −5.10852886727198138746625955152, −3.91368648437557041470435138269, −3.23627008718647958502903040571, −2.71994668641778708898352781098, −2.09635003626579831233533076903, −0.97244220080250420168455235317, −0.08751626659078968835138025402, 1.30190398263701835733007030496, 2.20044321671203055512166666407, 3.12606294682908203717535597265, 4.021798466761440312905494944061, 4.73773979391415021071665550209, 5.297750636339770490304184026023, 6.2088178036479035748248776305, 6.6430469129647041766558606552, 7.153549083760332844809095980137, 8.24406092322904516738789154141, 8.76719017303667811991685200693, 9.52346502446648618455854775556, 9.969490242549097414860585648240, 10.59014207236895669726716504236, 11.94016564557591251482032022300, 12.58180325611607295067356171937, 13.042011136228620443077032122180, 13.57125615030454164876696175324, 14.28266168244234016282707238907, 14.97119368281922611900812777134, 15.557722055385420587841761417, 16.433830698409620446592830391322, 16.69037515541958788441289067162, 17.41340976707784371708932007077, 17.92127980882546710995575101253

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