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Properties

Label 1-6003-6003.1051-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.769 - 0.639i$
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  + (0.440 − 0.897i)2-s + (−0.612 − 0.790i)4-s + (0.704 + 0.709i)5-s + (−0.195 − 0.980i)7-s + (−0.979 + 0.202i)8-s + (0.947 − 0.320i)10-s + (−0.568 − 0.822i)11-s + (0.644 + 0.764i)13-s + (−0.966 − 0.255i)14-s + (−0.248 + 0.968i)16-s + (−0.959 − 0.281i)17-s + (−0.377 + 0.925i)19-s + (0.128 − 0.991i)20-s + (−0.988 + 0.149i)22-s + (−0.00679 + 0.999i)25-s + (0.970 − 0.242i)26-s + ⋯
L(s)  = 1  + (0.440 − 0.897i)2-s + (−0.612 − 0.790i)4-s + (0.704 + 0.709i)5-s + (−0.195 − 0.980i)7-s + (−0.979 + 0.202i)8-s + (0.947 − 0.320i)10-s + (−0.568 − 0.822i)11-s + (0.644 + 0.764i)13-s + (−0.966 − 0.255i)14-s + (−0.248 + 0.968i)16-s + (−0.959 − 0.281i)17-s + (−0.377 + 0.925i)19-s + (0.128 − 0.991i)20-s + (−0.988 + 0.149i)22-s + (−0.00679 + 0.999i)25-s + (0.970 − 0.242i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.769 - 0.639i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.769 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.806335602 - 0.6526053127i\)
\(L(\frac12)\) \(\approx\) \(1.806335602 - 0.6526053127i\)
\(L(1)\) \(\approx\) \(1.131688616 - 0.5365690467i\)
\(L(1)\) \(\approx\) \(1.131688616 - 0.5365690467i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.440 - 0.897i)T \)
5 \( 1 + (0.704 + 0.709i)T \)
7 \( 1 + (-0.195 - 0.980i)T \)
11 \( 1 + (-0.568 - 0.822i)T \)
13 \( 1 + (0.644 + 0.764i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (-0.377 + 0.925i)T \)
31 \( 1 + (0.938 - 0.346i)T \)
37 \( 1 + (0.882 - 0.470i)T \)
41 \( 1 + (-0.327 - 0.945i)T \)
43 \( 1 + (0.938 + 0.346i)T \)
47 \( 1 + (-0.733 + 0.680i)T \)
53 \( 1 + (-0.523 + 0.852i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (0.998 - 0.0543i)T \)
67 \( 1 + (-0.568 + 0.822i)T \)
71 \( 1 + (-0.933 + 0.359i)T \)
73 \( 1 + (0.986 + 0.162i)T \)
79 \( 1 + (0.963 + 0.268i)T \)
83 \( 1 + (0.534 + 0.844i)T \)
89 \( 1 + (0.0203 + 0.999i)T \)
97 \( 1 + (0.760 - 0.649i)T \)
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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

−17.74234433353582665409980091186, −17.23102984653211811521482924519, −16.24667324351131096230863663532, −15.81332480814333567962038941471, −15.182159416201642182607094403014, −14.7979240132391023332816384648, −13.648268013404388103877660828422, −13.22222924846398958493890037512, −12.791322931674061084278673554178, −12.19025674707419708773105856579, −11.337481646292542098005354649, −10.334059894550068885221635335900, −9.54971323522057994811129850396, −9.0010386969329112751985667721, −8.35704723659951983929579112526, −7.87806303760295756558913489318, −6.70245483316441780926637038466, −6.28806068105456186614319739730, −5.61661860829845086007700595521, −4.77213846593653279047242560212, −4.6324366873021131696428922378, −3.33228041476112841926338799963, −2.58118775490103446039564230253, −1.87925963380689918035529556833, −0.51356723302037697135176808773, 0.785423376017944056457428360630, 1.62657327918419948804577822689, 2.44233006816540368231676046720, 3.06763120354541153301179382382, 3.91672121304113181250673344857, 4.37318118759744988124537139391, 5.402325095942284670586459985573, 6.21121829992782865803135402724, 6.48822312284321985109467067503, 7.539066073523596221407472877399, 8.43292474733431382579430954790, 9.2616588407287902541056823462, 9.822741265479774658079934460059, 10.63088693087523893931482061221, 10.90769275278653190972805159623, 11.47433125399322517729763595040, 12.467692464689531739216924191782, 13.283135831549205414763198467109, 13.6085000874099832510618931907, 14.1151344293867377475096949283, 14.69974431102515384045760746711, 15.64222038442404694885064627114, 16.30332969365517616526924217173, 17.153936657825384684211726180836, 17.79755284958189440492123930689

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