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Properties

Label 1-4003-4003.3238-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.681 - 0.731i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (0.935 + 0.353i)2-s + (0.837 − 0.546i)3-s + (0.750 + 0.661i)4-s + (−0.968 − 0.250i)5-s + (0.976 − 0.214i)6-s + (−0.619 − 0.785i)7-s + (0.468 + 0.883i)8-s + (0.403 − 0.915i)9-s + (−0.817 − 0.576i)10-s + (0.796 − 0.605i)11-s + (0.989 + 0.143i)12-s + (−0.561 + 0.827i)13-s + (−0.302 − 0.953i)14-s + (−0.947 + 0.319i)15-s + (0.126 + 0.992i)16-s + (0.647 − 0.762i)17-s + ⋯
L(s)  = 1  + (0.935 + 0.353i)2-s + (0.837 − 0.546i)3-s + (0.750 + 0.661i)4-s + (−0.968 − 0.250i)5-s + (0.976 − 0.214i)6-s + (−0.619 − 0.785i)7-s + (0.468 + 0.883i)8-s + (0.403 − 0.915i)9-s + (−0.817 − 0.576i)10-s + (0.796 − 0.605i)11-s + (0.989 + 0.143i)12-s + (−0.561 + 0.827i)13-s + (−0.302 − 0.953i)14-s + (−0.947 + 0.319i)15-s + (0.126 + 0.992i)16-s + (0.647 − 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.681 - 0.731i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (3238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.681 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.401272387 - 1.479945865i\)
\(L(\frac12)\) \(\approx\) \(3.401272387 - 1.479945865i\)
\(L(1)\) \(\approx\) \(2.081585190 - 0.2559570584i\)
\(L(1)\) \(\approx\) \(2.081585190 - 0.2559570584i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.935 + 0.353i)T \)
3 \( 1 + (0.837 - 0.546i)T \)
5 \( 1 + (-0.968 - 0.250i)T \)
7 \( 1 + (-0.619 - 0.785i)T \)
11 \( 1 + (0.796 - 0.605i)T \)
13 \( 1 + (-0.561 + 0.827i)T \)
17 \( 1 + (0.647 - 0.762i)T \)
19 \( 1 + (0.336 + 0.941i)T \)
23 \( 1 + (0.126 + 0.992i)T \)
29 \( 1 + (0.336 - 0.941i)T \)
31 \( 1 + (0.197 - 0.980i)T \)
37 \( 1 + (0.336 - 0.941i)T \)
41 \( 1 + (0.796 + 0.605i)T \)
43 \( 1 + (-0.891 + 0.452i)T \)
47 \( 1 + (0.126 - 0.992i)T \)
53 \( 1 + (0.126 - 0.992i)T \)
59 \( 1 + (0.530 - 0.847i)T \)
61 \( 1 + (-0.674 + 0.738i)T \)
67 \( 1 + (-0.302 + 0.953i)T \)
71 \( 1 + (0.837 - 0.546i)T \)
73 \( 1 + (-0.856 - 0.515i)T \)
79 \( 1 + (-0.674 - 0.738i)T \)
83 \( 1 + (0.935 + 0.353i)T \)
89 \( 1 + (-0.302 - 0.953i)T \)
97 \( 1 + (-0.302 + 0.953i)T \)
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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

−18.99059599748584809358282266321, −18.17011320490900938205752554630, −16.89576336095993201669058042019, −16.191696574444406663963517370050, −15.455374385713952526690918445511, −15.18326865610827256937935570303, −14.575713251895145897150958475061, −14.01393949131291530901435555474, −12.91085497063605877625072853652, −12.44530073575916714079159247256, −11.99023752714494142631063923206, −10.95688663994921794514101166394, −10.384185412938152775378354676329, −9.696234990410001360162698946228, −8.89490068491994255499048571624, −8.14683233944676524798314072687, −7.20056102617244801592705232593, −6.710590392711873579563533288085, −5.64917304779093497553904930421, −4.76879661905140689515425751373, −4.31103886426885922407380412511, −3.23044017023928206995087421970, −3.09728798280895482807707977034, −2.25040110524776640923239978888, −1.09625523935918020664075347516, 0.71839986250002607777784148870, 1.696125391985389248643276725908, 2.796788761025414701937688012914, 3.5397804471111078238011050832, 3.90130985195059002348688161906, 4.61688790806590778145388413181, 5.80121231980172259581710819670, 6.5488821084926687750627192257, 7.279160302824852641441614813823, 7.64597032734879172764356488425, 8.33778022790320266181601740435, 9.31756605016479488747457342571, 9.913216207651413069670885458770, 11.248833563091734546317134528272, 11.84252217248962169542202354009, 12.18561503096327070802021053668, 13.23269954721432002777734327102, 13.52088449894583908746133270733, 14.45224614422764690705778784760, 14.62945042305475001235213512603, 15.63084109294101660205677123511, 16.358743285171264392981442116972, 16.63928039719454304124018411922, 17.53192293352907221489827169834, 18.70059425538278768653480823594

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