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Properties

Label 1-6003-6003.2459-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: $\chi_{6003} (2459, \cdot )$
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

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

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