↑

Properties

Label 1-6003-6003.1024-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.246 - 0.969i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.546 − 0.837i)2-s + (−0.403 − 0.915i)4-s + (0.810 + 0.585i)5-s + (0.155 + 0.987i)7-s + (−0.986 − 0.162i)8-s + (0.933 − 0.359i)10-s + (0.169 − 0.985i)11-s + (−0.997 + 0.0679i)13-s + (0.912 + 0.409i)14-s + (−0.675 + 0.737i)16-s + (0.654 − 0.755i)17-s + (0.591 − 0.806i)19-s + (0.209 − 0.977i)20-s + (−0.733 − 0.680i)22-s + (0.314 + 0.949i)25-s + (−0.488 + 0.872i)26-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)2-s + (−0.403 − 0.915i)4-s + (0.810 + 0.585i)5-s + (0.155 + 0.987i)7-s + (−0.986 − 0.162i)8-s + (0.933 − 0.359i)10-s + (0.169 − 0.985i)11-s + (−0.997 + 0.0679i)13-s + (0.912 + 0.409i)14-s + (−0.675 + 0.737i)16-s + (0.654 − 0.755i)17-s + (0.591 − 0.806i)19-s + (0.209 − 0.977i)20-s + (−0.733 − 0.680i)22-s + (0.314 + 0.949i)25-s + (−0.488 + 0.872i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.246 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.170978162 - 1.688426457i\)
\(L(\frac12)\) \(\approx\) \(2.170978162 - 1.688426457i\)
\(L(1)\) \(\approx\) \(1.420441069 - 0.6304626143i\)
\(L(1)\) \(\approx\) \(1.420441069 - 0.6304626143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.546 - 0.837i)T \)
5 \( 1 + (0.810 + 0.585i)T \)
7 \( 1 + (0.155 + 0.987i)T \)
11 \( 1 + (0.169 - 0.985i)T \)
13 \( 1 + (-0.997 + 0.0679i)T \)
17 \( 1 + (0.654 - 0.755i)T \)
19 \( 1 + (0.591 - 0.806i)T \)
31 \( 1 + (0.612 + 0.790i)T \)
37 \( 1 + (0.523 + 0.852i)T \)
41 \( 1 + (-0.0475 - 0.998i)T \)
43 \( 1 + (0.612 - 0.790i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (0.685 + 0.728i)T \)
59 \( 1 + (0.928 + 0.371i)T \)
61 \( 1 + (0.833 - 0.552i)T \)
67 \( 1 + (-0.169 - 0.985i)T \)
71 \( 1 + (0.0203 + 0.999i)T \)
73 \( 1 + (-0.182 + 0.983i)T \)
79 \( 1 + (-0.976 + 0.215i)T \)
83 \( 1 + (-0.984 + 0.175i)T \)
89 \( 1 + (0.818 - 0.574i)T \)
97 \( 1 + (0.248 - 0.968i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.67528112038975860331471884228, −17.111876591849826735443258812924, −16.532152080471061473955603463666, −16.135670649794902487367942999787, −14.95940238904062220884374901203, −14.54105239357495696907482016577, −14.143423567293154699345838490115, −13.10119647682940676097576101462, −12.95600774535064530538414272980, −12.153690849681986007512942499359, −11.50851293795149353592580361993, −10.213019324200232879872022530498, −9.87394029153105062162839793317, −9.2611038893069438248451815854, −8.179634335718449880680327634951, −7.73628684687229964187259961818, −7.09648146402986308102689512168, −6.290148503913896808180193868247, −5.66762557435857247793600436507, −4.88081938075085595513457586995, −4.39757674558187057585390567650, −3.697979959605154027083930757716, −2.6700143542651004396580762614, −1.79518400972360780364483879807, −0.854184294432857232372188057678, 0.689681955372063890218904312666, 1.62645822521933933819552116299, 2.6040713631619042903923724791, 2.76964507937383245645050495504, 3.57826560180264165053763029708, 4.75761812708622752892196068985, 5.353038523953410620090866720934, 5.75616566558331813811533063256, 6.61952604299027336314879525354, 7.30277080069492219513449922130, 8.57031273339551952144264198931, 9.03264071869527113674809391776, 9.87264811847638688665943374291, 10.146419030073911787729620368913, 11.24934663518803089153406966223, 11.54315978707820520672571439426, 12.25235424198653460287676087531, 12.93987750249787682938526745083, 13.83042711324676831847078573475, 14.06057313190308442113841490461, 14.76473515420445950358222153910, 15.40950430290950361091290476411, 16.11469144045672792117811585518, 17.14084513708478790155546209955, 17.73688959376976055198329017446

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