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Properties

Label 1-6003-6003.1184-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.334 + 0.942i$
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  + (0.923 − 0.384i)2-s + (0.704 − 0.709i)4-s + (0.644 − 0.764i)5-s + (−0.976 − 0.215i)7-s + (0.377 − 0.925i)8-s + (0.301 − 0.953i)10-s + (−0.546 − 0.837i)11-s + (−0.973 − 0.229i)13-s + (−0.984 + 0.175i)14-s + (−0.00679 − 0.999i)16-s + (−0.654 + 0.755i)17-s + (−0.262 − 0.965i)19-s + (−0.0882 − 0.996i)20-s + (−0.826 − 0.563i)22-s + (−0.169 − 0.985i)25-s + (−0.986 + 0.162i)26-s + ⋯
L(s)  = 1  + (0.923 − 0.384i)2-s + (0.704 − 0.709i)4-s + (0.644 − 0.764i)5-s + (−0.976 − 0.215i)7-s + (0.377 − 0.925i)8-s + (0.301 − 0.953i)10-s + (−0.546 − 0.837i)11-s + (−0.973 − 0.229i)13-s + (−0.984 + 0.175i)14-s + (−0.00679 − 0.999i)16-s + (−0.654 + 0.755i)17-s + (−0.262 − 0.965i)19-s + (−0.0882 − 0.996i)20-s + (−0.826 − 0.563i)22-s + (−0.169 − 0.985i)25-s + (−0.986 + 0.162i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.334 + 0.942i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.334 + 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5331335236 - 0.7549193479i\)
\(L(\frac12)\) \(\approx\) \(-0.5331335236 - 0.7549193479i\)
\(L(1)\) \(\approx\) \(1.127378742 - 0.8113203545i\)
\(L(1)\) \(\approx\) \(1.127378742 - 0.8113203545i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.923 - 0.384i)T \)
5 \( 1 + (0.644 - 0.764i)T \)
7 \( 1 + (-0.976 - 0.215i)T \)
11 \( 1 + (-0.546 - 0.837i)T \)
13 \( 1 + (-0.973 - 0.229i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (-0.262 - 0.965i)T \)
31 \( 1 + (-0.833 + 0.552i)T \)
37 \( 1 + (-0.947 + 0.320i)T \)
41 \( 1 + (0.888 + 0.458i)T \)
43 \( 1 + (0.833 + 0.552i)T \)
47 \( 1 + (0.988 - 0.149i)T \)
53 \( 1 + (-0.933 - 0.359i)T \)
59 \( 1 + (0.786 - 0.618i)T \)
61 \( 1 + (-0.209 - 0.977i)T \)
67 \( 1 + (0.546 - 0.837i)T \)
71 \( 1 + (-0.970 + 0.242i)T \)
73 \( 1 + (-0.591 + 0.806i)T \)
79 \( 1 + (-0.869 + 0.494i)T \)
83 \( 1 + (0.999 - 0.0271i)T \)
89 \( 1 + (0.488 - 0.872i)T \)
97 \( 1 + (-0.390 + 0.920i)T \)
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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

−17.93315203706085924976436147723, −17.51400901488285964900061798867, −16.72612229678650290433259020740, −16.06203028707872355811286228470, −15.46491372236072497678434885531, −14.80299143336597412294092626189, −14.32167387975995739442725954170, −13.57924317268152943466574564999, −13.02161145319096121089223603234, −12.34721267627603909589063536785, −11.90326017369227872079020560140, −10.80296892571876178764546773355, −10.36990367747788344193370787715, −9.53333909989787436982800413352, −8.983220635705303488445403246688, −7.71779402025479134888764889634, −7.14471410867760865445362285277, −6.8204372102466038213723381638, −5.76583812631565905690683033195, −5.57815377416696228455079754674, −4.49983364477167622488605925335, −3.86062242901489288911092989206, −2.880712509615700317921318413401, −2.445904251109431973657328315391, −1.84059540216744684791671178442, 0.148611285316423889259892850691, 1.06560656805844892020005943078, 2.127641101513068874307385279950, 2.68306163711609984196820408912, 3.4499717756222364194254281556, 4.29376868998102433323849498893, 4.99400445570319871356119819662, 5.59906074415821294313100713812, 6.275323230532370863582143567740, 6.85916418395820207740926807713, 7.75067676640439323390717806876, 8.78177305500066829205943483572, 9.36957868895714847442743266347, 10.08628919586659800688031504623, 10.68309504668276431136196121066, 11.29602154779757689182710074843, 12.361959897173846621217441056442, 12.77538474800386055913308325144, 13.153446941996044180569907303661, 13.83655603266492778008197315642, 14.42513411119845555940972320024, 15.34699869917199863207113227082, 15.91827953061375402855701011632, 16.40282399726978841577590882630, 17.22113768434473027812253550906

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