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Properties

Label 1-6003-6003.2812-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.913 - 0.407i$
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  + (−0.981 + 0.189i)2-s + (0.928 − 0.371i)4-s + (−0.995 + 0.0950i)5-s + (0.723 − 0.690i)7-s + (−0.841 + 0.540i)8-s + (0.959 − 0.281i)10-s + (0.327 + 0.945i)11-s + (0.723 + 0.690i)13-s + (−0.580 + 0.814i)14-s + (0.723 − 0.690i)16-s + (0.142 + 0.989i)17-s + (0.142 − 0.989i)19-s + (−0.888 + 0.458i)20-s + (−0.5 − 0.866i)22-s + (0.981 − 0.189i)25-s + (−0.841 − 0.540i)26-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)2-s + (0.928 − 0.371i)4-s + (−0.995 + 0.0950i)5-s + (0.723 − 0.690i)7-s + (−0.841 + 0.540i)8-s + (0.959 − 0.281i)10-s + (0.327 + 0.945i)11-s + (0.723 + 0.690i)13-s + (−0.580 + 0.814i)14-s + (0.723 − 0.690i)16-s + (0.142 + 0.989i)17-s + (0.142 − 0.989i)19-s + (−0.888 + 0.458i)20-s + (−0.5 − 0.866i)22-s + (0.981 − 0.189i)25-s + (−0.841 − 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.913 - 0.407i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (2812, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.913 - 0.407i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076080055 - 0.2289118995i\)
\(L(\frac12)\) \(\approx\) \(1.076080055 - 0.2289118995i\)
\(L(1)\) \(\approx\) \(0.7266327722 + 0.01956907393i\)
\(L(1)\) \(\approx\) \(0.7266327722 + 0.01956907393i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.981 + 0.189i)T \)
5 \( 1 + (-0.995 + 0.0950i)T \)
7 \( 1 + (0.723 - 0.690i)T \)
11 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (0.723 + 0.690i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (0.888 + 0.458i)T \)
37 \( 1 + (-0.415 - 0.909i)T \)
41 \( 1 + (0.995 - 0.0950i)T \)
43 \( 1 + (0.888 - 0.458i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.723 + 0.690i)T \)
61 \( 1 + (-0.0475 - 0.998i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.995 - 0.0950i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.580 - 0.814i)T \)
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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

−17.86855795635742396336340407708, −17.1762959065050741187022782403, −16.39313811071075185191390146997, −15.83682537925446777110172459958, −15.501639480831526342596146837608, −14.60588030426341623314624177199, −13.99419077331448620961560529877, −12.87453447882540252789523920994, −12.239519119863395719051355178602, −11.54752065700804220437615338282, −11.2916667520181192798212495751, −10.577932108504385206782539048118, −9.71139288259661556847878156735, −8.8759918525584045153804346490, −8.439724720797131339562512564675, −7.854528313352370340423304970380, −7.387942989621027861184523842422, −6.230568835810823715335653723603, −5.80515171467655897252398226973, −4.768255625083631251247056980764, −3.844029992487768755625314399098, −3.104838015808861308710951034153, −2.54229020374787577517610972026, −1.26388342315683206655436910538, −0.85766202001484537190328090784, 0.54313346216230496481090839754, 1.39643769359015397958444670251, 2.06664531287714004444600615737, 3.132042195776062165793363754541, 4.09173484698870196228999385989, 4.49614294636326997853791387975, 5.56560831925979865161640629923, 6.545780908798027011194806055455, 7.13523069060592275141495833166, 7.53563870521300868178543109350, 8.403217036041149144686673025661, 8.78653966128657907883723186500, 9.643183018406231893455889347581, 10.52425760280697899869438333238, 10.92842854015029718255441970931, 11.55894195425364808290849079882, 12.1451760520326110142219230917, 12.91531045780069121382199162072, 14.07002041339730695871944230316, 14.49614122399657488110487812901, 15.32216820413595463212542355423, 15.680526608645501523474381430956, 16.44883068643459511266875369841, 17.12776287404279881991445756041, 17.652596834421624979730578682184

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