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Properties

Label 1-6003-6003.1001-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.533 - 0.845i$
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  + (0.949 − 0.314i)2-s + (0.802 − 0.596i)4-s + (0.963 − 0.268i)5-s + (−0.997 − 0.0679i)7-s + (0.574 − 0.818i)8-s + (0.830 − 0.557i)10-s + (−0.961 + 0.275i)11-s + (−0.869 + 0.494i)13-s + (−0.968 + 0.248i)14-s + (0.288 − 0.957i)16-s + (0.755 + 0.654i)17-s + (−0.396 + 0.917i)19-s + (0.612 − 0.790i)20-s + (−0.826 + 0.563i)22-s + (0.855 − 0.517i)25-s + (−0.670 + 0.742i)26-s + ⋯
L(s)  = 1  + (0.949 − 0.314i)2-s + (0.802 − 0.596i)4-s + (0.963 − 0.268i)5-s + (−0.997 − 0.0679i)7-s + (0.574 − 0.818i)8-s + (0.830 − 0.557i)10-s + (−0.961 + 0.275i)11-s + (−0.869 + 0.494i)13-s + (−0.968 + 0.248i)14-s + (0.288 − 0.957i)16-s + (0.755 + 0.654i)17-s + (−0.396 + 0.917i)19-s + (0.612 − 0.790i)20-s + (−0.826 + 0.563i)22-s + (0.855 − 0.517i)25-s + (−0.670 + 0.742i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.533 - 0.845i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.533 - 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248604379 - 2.263748327i\)
\(L(\frac12)\) \(\approx\) \(1.248604379 - 2.263748327i\)
\(L(1)\) \(\approx\) \(1.616807926 - 0.6221902119i\)
\(L(1)\) \(\approx\) \(1.616807926 - 0.6221902119i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.949 - 0.314i)T \)
5 \( 1 + (0.963 - 0.268i)T \)
7 \( 1 + (-0.997 - 0.0679i)T \)
11 \( 1 + (-0.961 + 0.275i)T \)
13 \( 1 + (-0.869 + 0.494i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.396 + 0.917i)T \)
31 \( 1 + (0.0135 - 0.999i)T \)
37 \( 1 + (0.994 - 0.101i)T \)
41 \( 1 + (-0.998 + 0.0475i)T \)
43 \( 1 + (-0.0135 - 0.999i)T \)
47 \( 1 + (0.149 - 0.988i)T \)
53 \( 1 + (-0.996 + 0.0815i)T \)
59 \( 1 + (-0.928 - 0.371i)T \)
61 \( 1 + (0.935 - 0.352i)T \)
67 \( 1 + (0.275 - 0.961i)T \)
71 \( 1 + (-0.452 - 0.891i)T \)
73 \( 1 + (-0.470 - 0.882i)T \)
79 \( 1 + (0.229 - 0.973i)T \)
83 \( 1 + (-0.390 - 0.920i)T \)
89 \( 1 + (0.998 + 0.0611i)T \)
97 \( 1 + (0.0271 - 0.999i)T \)
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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

−17.73840091995917672543019254244, −17.214703099198460689179609356612, −16.435011630990905526827863863336, −15.96167414521468960762653575222, −15.26620860698626231464033113336, −14.57920081778130960747226028316, −13.98242421621596361864130117827, −13.26447982288239972936634687638, −12.88587496258815157035689463753, −12.34525700967146159886949139429, −11.39515885025771689719788958599, −10.66124558886806241211775429597, −10.02242433126546264668413927209, −9.459680809768165422874759933804, −8.47697960374272346093424441090, −7.62820129445451479897045174914, −6.96684016293697958233043114993, −6.41128887261146241959293970430, −5.609210028412801954490945979433, −5.20697880961014102536388274551, −4.4498512040242297541471702196, −3.23294979411165065297630479729, −2.804468037784833206919121385469, −2.40919748544783932555810295700, −1.11651912767323710940406751388, 0.411700716219794095801072256617, 1.74769007083631689119117108674, 2.14712170788331385079083232022, 2.99941759864792023550130547378, 3.6819147524619936984496173177, 4.59806761493170132865196673962, 5.21406251628509056005118061107, 6.00031398303546306333024731853, 6.31233669240526353614789759318, 7.256378101477685928114591930761, 7.92037526244905169140474632295, 9.07171345038018695854997903412, 9.88356302184220961205764616922, 10.11737268328294549798183367380, 10.73577560775314619512402803239, 11.85915531498132984354440180664, 12.43121685515975628577388805237, 12.90682540515070628631737048373, 13.440971640654619345947492928584, 14.094211129816747876342193709124, 14.8039904981802499548275720981, 15.32995835198067196978964251135, 16.24330176786031990381070020046, 16.74660992571709042194783231578, 17.20111827804167955024964318739

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