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Properties

Label 1-6003-6003.1094-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.0681 - 0.997i$
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  + (−0.229 − 0.973i)2-s + (−0.894 + 0.446i)4-s + (0.534 + 0.844i)5-s + (−0.248 − 0.968i)7-s + (0.639 + 0.768i)8-s + (0.699 − 0.714i)10-s + (−0.0679 − 0.997i)11-s + (−0.999 − 0.0271i)13-s + (−0.885 + 0.464i)14-s + (0.601 − 0.798i)16-s + (−0.281 − 0.959i)17-s + (0.998 + 0.0611i)19-s + (−0.855 − 0.517i)20-s + (−0.955 + 0.294i)22-s + (−0.427 + 0.903i)25-s + (0.202 + 0.979i)26-s + ⋯
L(s)  = 1  + (−0.229 − 0.973i)2-s + (−0.894 + 0.446i)4-s + (0.534 + 0.844i)5-s + (−0.248 − 0.968i)7-s + (0.639 + 0.768i)8-s + (0.699 − 0.714i)10-s + (−0.0679 − 0.997i)11-s + (−0.999 − 0.0271i)13-s + (−0.885 + 0.464i)14-s + (0.601 − 0.798i)16-s + (−0.281 − 0.959i)17-s + (0.998 + 0.0611i)19-s + (−0.855 − 0.517i)20-s + (−0.955 + 0.294i)22-s + (−0.427 + 0.903i)25-s + (0.202 + 0.979i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0681 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0681 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.0681 - 0.997i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.0681 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9891205555 - 0.9238140965i\)
\(L(\frac12)\) \(\approx\) \(0.9891205555 - 0.9238140965i\)
\(L(1)\) \(\approx\) \(0.8135603413 - 0.4038883129i\)
\(L(1)\) \(\approx\) \(0.8135603413 - 0.4038883129i\)

Euler product

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

−17.64867531860656602129791360404, −17.287533813944823197604837664204, −16.60913530682462506380114741148, −15.75021169352893973048682291591, −15.52333121563030430392473492099, −14.58870708783894293041862149751, −14.24150745567292940505950043660, −13.142577486171192184963175339047, −12.77431250188817517447098484914, −12.24178536442399130306081032758, −11.33901094859986705560744801539, −10.02223907470151210421167352767, −9.78665210041985577136532016229, −9.22791896518935212030928562327, −8.45853671573246345622805139929, −7.90843915697863905318198448833, −7.085811635380402012288250928179, −6.33936407819742680746434789873, −5.68633161392057558515857530445, −5.04125393087319390026580209997, −4.602319646061312590990986216323, −3.62889194652719557718434422032, −2.33826734425787478663098928428, −1.82051161006167592535937989463, −0.67796505682075764848373592964, 0.56633193429240280039933127155, 1.34006851314007053026424238723, 2.39070522146159858146177455935, 3.10047994617517395954395276585, 3.37178003230977490023141713366, 4.5407893462319074203919336720, 5.074219535751052577334123441587, 6.071213940198388728914057602726, 6.895824918665679631167341825851, 7.53904215302281864798132898375, 8.18994125616736407600101435414, 9.22674015030008382944239871121, 9.8294816333576850699371059192, 10.15199589042848390269930328671, 10.99269810870855291347967058121, 11.4531731449344681935290341250, 12.080762418308683657860123097961, 13.25084847647483673556648933476, 13.398526573819899042297050278602, 14.25027640570601968294060340926, 14.46450747181834996298760993323, 15.70270947680429645750288796594, 16.517020697977781361981786080117, 16.98832168888693713142295206295, 17.882810187328214154776662736116

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