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Properties

Label 1-6003-6003.104-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.947 - 0.318i$
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  + (−0.814 + 0.580i)2-s + (0.327 − 0.945i)4-s + (−0.888 − 0.458i)5-s + (−0.786 − 0.618i)7-s + (0.281 + 0.959i)8-s + (0.989 − 0.142i)10-s + (0.0950 − 0.995i)11-s + (0.786 − 0.618i)13-s + (0.998 + 0.0475i)14-s + (−0.786 − 0.618i)16-s + (−0.755 − 0.654i)17-s + (−0.755 + 0.654i)19-s + (−0.723 + 0.690i)20-s + (0.5 + 0.866i)22-s + (0.580 + 0.814i)25-s + (−0.281 + 0.959i)26-s + ⋯
L(s)  = 1  + (−0.814 + 0.580i)2-s + (0.327 − 0.945i)4-s + (−0.888 − 0.458i)5-s + (−0.786 − 0.618i)7-s + (0.281 + 0.959i)8-s + (0.989 − 0.142i)10-s + (0.0950 − 0.995i)11-s + (0.786 − 0.618i)13-s + (0.998 + 0.0475i)14-s + (−0.786 − 0.618i)16-s + (−0.755 − 0.654i)17-s + (−0.755 + 0.654i)19-s + (−0.723 + 0.690i)20-s + (0.5 + 0.866i)22-s + (0.580 + 0.814i)25-s + (−0.281 + 0.959i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.947 - 0.318i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.947 - 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08024511977 - 0.4900839984i\)
\(L(\frac12)\) \(\approx\) \(0.08024511977 - 0.4900839984i\)
\(L(1)\) \(\approx\) \(0.5288281943 - 0.1067446456i\)
\(L(1)\) \(\approx\) \(0.5288281943 - 0.1067446456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.814 + 0.580i)T \)
5 \( 1 + (-0.888 - 0.458i)T \)
7 \( 1 + (-0.786 - 0.618i)T \)
11 \( 1 + (0.0950 - 0.995i)T \)
13 \( 1 + (0.786 - 0.618i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (-0.755 + 0.654i)T \)
31 \( 1 + (0.690 - 0.723i)T \)
37 \( 1 + (-0.540 + 0.841i)T \)
41 \( 1 + (-0.458 + 0.888i)T \)
43 \( 1 + (-0.690 - 0.723i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (0.786 - 0.618i)T \)
61 \( 1 + (-0.971 - 0.235i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (0.371 - 0.928i)T \)
83 \( 1 + (0.888 - 0.458i)T \)
89 \( 1 + (0.281 - 0.959i)T \)
97 \( 1 + (-0.998 + 0.0475i)T \)
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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

−18.11611802193606030189782665767, −17.525960214050189999782909565115, −16.7541439202357149361127042264, −16.04150777261074419496490203146, −15.34986745561456891435928596025, −15.24038046691818491531293317386, −13.98725867647309198020284548181, −13.16222162737773249826861301047, −12.448485913170808286035487277111, −12.10935607904003486826449274283, −11.29922488310325197402704806363, −10.73169468775637712496260072514, −10.144057068987005550762592567959, −9.26755485292077270602746768132, −8.71726694020053065434497000396, −8.2477465547026304682973607201, −7.14977857780858552321142320937, −6.80880613964179502721410856850, −6.196738120817106356452819482252, −4.84423694593262629254191132608, −3.95017625915312823972196184095, −3.64477830347301955940768046875, −2.50826825242063322026823400644, −2.1826324647055042005503224255, −1.01395241678175134049823570237, 0.268072130062809248522316306105, 0.73733484893429961476902814343, 1.75948324765310948555214484917, 2.99947931124510142473902403802, 3.624180872566179362773797065674, 4.48294489725779026852992116586, 5.28176465374546471490842394654, 6.24877531659668802952530859261, 6.55886085428781066345356533728, 7.471502023145691738894293635722, 8.171465394207264741916660254229, 8.55190099531839948345212502611, 9.27675565255778020261281464185, 10.0812894153992599651160872523, 10.738919402079334981159244175091, 11.28922718357733570429019911732, 11.97598633489306022778099724528, 12.98155762128934162270799837226, 13.5335373333378274064781737606, 14.1469696205723899344685234754, 15.3447672876750003975119573475, 15.424511665603456716822545703300, 16.30468045843254394290104095841, 16.62979362270754688709704909292, 17.207647451556634777700696065221

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