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Properties

Label 1-6003-6003.16-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.361 + 0.932i$
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  + (−0.973 − 0.229i)2-s + (0.894 + 0.446i)4-s + (0.534 − 0.844i)5-s + (−0.248 + 0.968i)7-s + (−0.768 − 0.639i)8-s + (−0.714 + 0.699i)10-s + (−0.997 − 0.0679i)11-s + (0.999 − 0.0271i)13-s + (0.464 − 0.885i)14-s + (0.601 + 0.798i)16-s + (−0.959 − 0.281i)17-s + (−0.0611 − 0.998i)19-s + (0.855 − 0.517i)20-s + (0.955 + 0.294i)22-s + (−0.427 − 0.903i)25-s + (−0.979 − 0.202i)26-s + ⋯
L(s)  = 1  + (−0.973 − 0.229i)2-s + (0.894 + 0.446i)4-s + (0.534 − 0.844i)5-s + (−0.248 + 0.968i)7-s + (−0.768 − 0.639i)8-s + (−0.714 + 0.699i)10-s + (−0.997 − 0.0679i)11-s + (0.999 − 0.0271i)13-s + (0.464 − 0.885i)14-s + (0.601 + 0.798i)16-s + (−0.959 − 0.281i)17-s + (−0.0611 − 0.998i)19-s + (0.855 − 0.517i)20-s + (0.955 + 0.294i)22-s + (−0.427 − 0.903i)25-s + (−0.979 − 0.202i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.361 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4711783036 + 0.3227771748i\)
\(L(\frac12)\) \(\approx\) \(0.4711783036 + 0.3227771748i\)
\(L(1)\) \(\approx\) \(0.6375187626 - 0.07451104395i\)
\(L(1)\) \(\approx\) \(0.6375187626 - 0.07451104395i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.973 - 0.229i)T \)
5 \( 1 + (0.534 - 0.844i)T \)
7 \( 1 + (-0.248 + 0.968i)T \)
11 \( 1 + (-0.997 - 0.0679i)T \)
13 \( 1 + (0.999 - 0.0271i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (-0.0611 - 0.998i)T \)
31 \( 1 + (-0.546 - 0.837i)T \)
37 \( 1 + (0.917 + 0.396i)T \)
41 \( 1 + (-0.327 - 0.945i)T \)
43 \( 1 + (-0.546 + 0.837i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.947 + 0.320i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (-0.923 - 0.384i)T \)
67 \( 1 + (-0.997 + 0.0679i)T \)
71 \( 1 + (-0.301 + 0.953i)T \)
73 \( 1 + (-0.377 + 0.925i)T \)
79 \( 1 + (0.390 + 0.920i)T \)
83 \( 1 + (-0.848 - 0.529i)T \)
89 \( 1 + (0.970 - 0.242i)T \)
97 \( 1 + (-0.403 + 0.915i)T \)
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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

−17.88046013172699434383454681476, −16.850394781534845443815493994922, −16.49736302743765591813752553865, −15.721849504413498514219243805125, −15.08647326780816413063434604900, −14.47325546627199916720280447362, −13.55483604685560827551841123400, −13.29140933505763529940901910877, −12.20705011020453017919876138352, −11.22927689313783363701580302274, −10.68889770743221148610228805423, −10.39595088567727619130920389, −9.74759437764046810376653564482, −8.90062026334746451425074209172, −8.19797800684929562254450540541, −7.47000427860673900395738395094, −6.944991257341547149347917251660, −6.19550192762840139479632257906, −5.774227681908183253277181267098, −4.68324574147042586912420951416, −3.57427258321670612915442850972, −3.02912073707573481796367168644, −2.001241310790314786326623814365, −1.49249575223536637029970754977, −0.23497371156714319045171577310, 0.8024795484204185480526115890, 1.73235187236991548522116256399, 2.48144213116804300185757406148, 2.94028748254794677283657368221, 4.14283327935837837474738293545, 5.00382708646716515884111237352, 5.866500848915529609975082969148, 6.25190059630107139818258023298, 7.22035417011919764106862298722, 8.08695826204459708966454502268, 8.64577376630779931555236325224, 9.15414230572448689328708966378, 9.63250707636807209548466735211, 10.53845158315853412579700221431, 11.15460261617439096328106624046, 11.772172311799473456537107948329, 12.58724504707325795082562346359, 13.20068481822130760761932753410, 13.504573005691204684394019886816, 14.87724404910883279412127620091, 15.59073584548394825864200681778, 15.90610419727679897542469158529, 16.52921769545866273053692790216, 17.34925250422496791649398054031, 17.908267353739466639553417607834

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