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Properties

Label 1-6003-6003.2477-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.812 - 0.583i$
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  + (0.618 + 0.786i)2-s + (−0.235 + 0.971i)4-s + (−0.327 − 0.945i)5-s + (−0.888 − 0.458i)7-s + (−0.909 + 0.415i)8-s + (0.540 − 0.841i)10-s + (−0.371 − 0.928i)11-s + (0.888 − 0.458i)13-s + (−0.189 − 0.981i)14-s + (−0.888 − 0.458i)16-s + (−0.281 + 0.959i)17-s + (−0.281 − 0.959i)19-s + (0.995 − 0.0950i)20-s + (0.5 − 0.866i)22-s + (−0.786 + 0.618i)25-s + (0.909 + 0.415i)26-s + ⋯
L(s)  = 1  + (0.618 + 0.786i)2-s + (−0.235 + 0.971i)4-s + (−0.327 − 0.945i)5-s + (−0.888 − 0.458i)7-s + (−0.909 + 0.415i)8-s + (0.540 − 0.841i)10-s + (−0.371 − 0.928i)11-s + (0.888 − 0.458i)13-s + (−0.189 − 0.981i)14-s + (−0.888 − 0.458i)16-s + (−0.281 + 0.959i)17-s + (−0.281 − 0.959i)19-s + (0.995 − 0.0950i)20-s + (0.5 − 0.866i)22-s + (−0.786 + 0.618i)25-s + (0.909 + 0.415i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.812 - 0.583i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (2477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.812 - 0.583i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1607578198 - 0.4994959951i\)
\(L(\frac12)\) \(\approx\) \(0.1607578198 - 0.4994959951i\)
\(L(1)\) \(\approx\) \(0.9900767557 + 0.1076873835i\)
\(L(1)\) \(\approx\) \(0.9900767557 + 0.1076873835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.618 + 0.786i)T \)
5 \( 1 + (-0.327 - 0.945i)T \)
7 \( 1 + (-0.888 - 0.458i)T \)
11 \( 1 + (-0.371 - 0.928i)T \)
13 \( 1 + (0.888 - 0.458i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (-0.281 - 0.959i)T \)
31 \( 1 + (0.0950 - 0.995i)T \)
37 \( 1 + (-0.755 + 0.654i)T \)
41 \( 1 + (0.945 - 0.327i)T \)
43 \( 1 + (-0.0950 - 0.995i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (0.888 - 0.458i)T \)
61 \( 1 + (0.814 + 0.580i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (0.998 + 0.0475i)T \)
83 \( 1 + (0.327 - 0.945i)T \)
89 \( 1 + (-0.909 - 0.415i)T \)
97 \( 1 + (0.189 - 0.981i)T \)
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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

−18.063491873620447837151355442074, −17.808264951274369168082017133291, −16.28847253986036735467263890492, −15.888907570472059335027417555151, −15.31288616807660595748643782541, −14.52680988349079897366281323193, −14.09125415488655915535361839041, −13.30011541781282207397737972559, −12.63033089385304660943904328456, −12.09782645557621451825118479788, −11.42792321336790468735622164128, −10.74498318040699231558551796892, −10.17732661375812822498112742550, −9.527903330967110462086522498203, −8.9165229449202210253983617169, −7.876248960579501022987603086226, −6.91962644573010701117905408829, −6.48937836240869479307915681122, −5.77189035022707892958806049231, −4.953980945905179974759296649270, −4.03566139704822417505373793977, −3.55691953021295312863036188100, −2.71375117393247447584870930894, −2.270325279846182240201706008128, −1.27071857542150100112321661676, 0.12319740812161828477918675147, 0.93378053678746802949876800318, 2.31974642335444881991817383253, 3.29762634211999073319878532628, 3.80854422548406770500519345272, 4.41495702930106647330914571874, 5.33367213590967644841141069310, 5.93231331826477549298276447573, 6.46858991304092827394642591205, 7.36386480032291205425121184828, 8.0292087057184746798785966096, 8.75169563997533726528664121858, 9.0174375461598472838422351613, 10.15841169624696967838635688971, 10.96039212576214634703893435038, 11.62769679328858657648848335003, 12.58835862913794236506504449939, 12.94045903069839577705092582282, 13.53709825366071331994884935983, 13.89100730260769641039551420404, 15.151902167545356731217151547902, 15.55757051335770444226040181550, 16.0356929817807760134614376264, 16.72036499184500445532630128760, 17.14905884386565508122098014964

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