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Properties

Label 1-6003-6003.1192-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.745 + 0.666i$
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  + (0.709 − 0.704i)2-s + (0.00679 − 0.999i)4-s + (−0.169 + 0.985i)5-s + (−0.906 + 0.421i)7-s + (−0.699 − 0.714i)8-s + (0.574 + 0.818i)10-s + (0.915 − 0.403i)11-s + (−0.894 + 0.446i)13-s + (−0.346 + 0.938i)14-s + (−0.999 − 0.0135i)16-s + (−0.989 − 0.142i)17-s + (0.505 − 0.862i)19-s + (0.984 + 0.175i)20-s + (0.365 − 0.930i)22-s + (−0.942 − 0.333i)25-s + (−0.320 + 0.947i)26-s + ⋯
L(s)  = 1  + (0.709 − 0.704i)2-s + (0.00679 − 0.999i)4-s + (−0.169 + 0.985i)5-s + (−0.906 + 0.421i)7-s + (−0.699 − 0.714i)8-s + (0.574 + 0.818i)10-s + (0.915 − 0.403i)11-s + (−0.894 + 0.446i)13-s + (−0.346 + 0.938i)14-s + (−0.999 − 0.0135i)16-s + (−0.989 − 0.142i)17-s + (0.505 − 0.862i)19-s + (0.984 + 0.175i)20-s + (0.365 − 0.930i)22-s + (−0.942 − 0.333i)25-s + (−0.320 + 0.947i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.745 + 0.666i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.745 + 0.666i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.254719071 + 0.4792394455i\)
\(L(\frac12)\) \(\approx\) \(1.254719071 + 0.4792394455i\)
\(L(1)\) \(\approx\) \(1.140411285 - 0.2640594358i\)
\(L(1)\) \(\approx\) \(1.140411285 - 0.2640594358i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.709 - 0.704i)T \)
5 \( 1 + (-0.169 + 0.985i)T \)
7 \( 1 + (-0.906 + 0.421i)T \)
11 \( 1 + (0.915 - 0.403i)T \)
13 \( 1 + (-0.894 + 0.446i)T \)
17 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (0.505 - 0.862i)T \)
31 \( 1 + (0.920 - 0.390i)T \)
37 \( 1 + (0.607 - 0.794i)T \)
41 \( 1 + (0.814 + 0.580i)T \)
43 \( 1 + (0.920 + 0.390i)T \)
47 \( 1 + (-0.294 + 0.955i)T \)
53 \( 1 + (-0.742 + 0.670i)T \)
59 \( 1 + (0.235 + 0.971i)T \)
61 \( 1 + (0.409 - 0.912i)T \)
67 \( 1 + (-0.403 + 0.915i)T \)
71 \( 1 + (-0.882 - 0.470i)T \)
73 \( 1 + (-0.953 - 0.301i)T \)
79 \( 1 + (-0.859 + 0.511i)T \)
83 \( 1 + (-0.998 - 0.0543i)T \)
89 \( 1 + (-0.852 - 0.523i)T \)
97 \( 1 + (0.719 + 0.694i)T \)
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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

−17.25866721722586102235777862767, −17.01245209871897800270026623663, −16.161849362834955538016922302914, −15.83238783524619303751203481866, −15.04932673928124963535303755266, −14.3768905775695686061701152669, −13.66074406574928492308989303631, −13.0611048263831609383535956935, −12.4765189577913702446808776555, −12.05033306410649454659696810, −11.3304255334866434699789702669, −10.12843306564185944843475115262, −9.566753827194697484124267876170, −8.85502641003592473317630808088, −8.188176268610693272467829148121, −7.38842599215112908631845485687, −6.854084719657444256559314833004, −6.08819328644906781073844028927, −5.44950661721490357590762539568, −4.52278398816252580121669846851, −4.197456193825154010922454360283, −3.38847232974562266125389547725, −2.55437289002993740195952356730, −1.49256338379890316517258979137, −0.307866568469563266513208565425, 0.84893418301805842934810272870, 2.03735193613668856607375323426, 2.87104247488224085823411045781, 2.956194198914342066513158622747, 4.2229188566202702065908152002, 4.4233639864455618625516588575, 5.74049856757940219254621615094, 6.19260290430420265372542116203, 6.85132701622047091471253926435, 7.39025773433150842413460002030, 8.727357760632018771906215623916, 9.54027923166381164490020684797, 9.667871592097877573904552526705, 10.751291389483006520983760363551, 11.30539733878724865221519136227, 11.77564488539196884341583311923, 12.47577215187034241963183035663, 13.22055069644058751459483665734, 13.82894855129722789496515908450, 14.496859611721922038275553448365, 14.942350131782132964902518893904, 15.7934410341236415690366307864, 16.11333200218556862236395623170, 17.323051025975229183879735900024, 17.89833233072459921526171562884

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