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Properties

Label 1-4003-4003.945-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.0948 - 0.995i$
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  + (−0.232 − 0.972i)2-s + (−0.968 + 0.250i)3-s + (−0.891 + 0.452i)4-s + (−0.302 + 0.953i)5-s + (0.468 + 0.883i)6-s + (0.197 − 0.980i)7-s + (0.647 + 0.762i)8-s + (0.874 − 0.484i)9-s + (0.997 + 0.0721i)10-s + (−0.994 − 0.108i)11-s + (0.750 − 0.661i)12-s + (−0.161 + 0.986i)13-s + (−0.999 + 0.0361i)14-s + (0.0541 − 0.998i)15-s + (0.590 − 0.806i)16-s + (−0.370 − 0.928i)17-s + ⋯
L(s)  = 1  + (−0.232 − 0.972i)2-s + (−0.968 + 0.250i)3-s + (−0.891 + 0.452i)4-s + (−0.302 + 0.953i)5-s + (0.468 + 0.883i)6-s + (0.197 − 0.980i)7-s + (0.647 + 0.762i)8-s + (0.874 − 0.484i)9-s + (0.997 + 0.0721i)10-s + (−0.994 − 0.108i)11-s + (0.750 − 0.661i)12-s + (−0.161 + 0.986i)13-s + (−0.999 + 0.0361i)14-s + (0.0541 − 0.998i)15-s + (0.590 − 0.806i)16-s + (−0.370 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0948 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0948 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.0948 - 0.995i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.0948 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5459753533 - 0.4964278255i\)
\(L(\frac12)\) \(\approx\) \(0.5459753533 - 0.4964278255i\)
\(L(1)\) \(\approx\) \(0.5802935372 - 0.2056590550i\)
\(L(1)\) \(\approx\) \(0.5802935372 - 0.2056590550i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.232 - 0.972i)T \)
3 \( 1 + (-0.968 + 0.250i)T \)
5 \( 1 + (-0.302 + 0.953i)T \)
7 \( 1 + (0.197 - 0.980i)T \)
11 \( 1 + (-0.994 - 0.108i)T \)
13 \( 1 + (-0.161 + 0.986i)T \)
17 \( 1 + (-0.370 - 0.928i)T \)
19 \( 1 + (0.989 + 0.143i)T \)
23 \( 1 + (0.590 - 0.806i)T \)
29 \( 1 + (0.989 - 0.143i)T \)
31 \( 1 + (0.837 + 0.546i)T \)
37 \( 1 + (0.989 - 0.143i)T \)
41 \( 1 + (-0.994 + 0.108i)T \)
43 \( 1 + (0.700 - 0.713i)T \)
47 \( 1 + (0.590 + 0.806i)T \)
53 \( 1 + (0.590 + 0.806i)T \)
59 \( 1 + (0.336 - 0.941i)T \)
61 \( 1 + (0.530 + 0.847i)T \)
67 \( 1 + (-0.999 - 0.0361i)T \)
71 \( 1 + (-0.968 + 0.250i)T \)
73 \( 1 + (0.907 + 0.419i)T \)
79 \( 1 + (0.530 - 0.847i)T \)
83 \( 1 + (-0.232 - 0.972i)T \)
89 \( 1 + (-0.999 + 0.0361i)T \)
97 \( 1 + (-0.999 - 0.0361i)T \)
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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

−18.21830601682640215957291128120, −17.9031956381767259251677050086, −17.31837456597746642960695188730, −16.580994869988347894235594819878, −15.93512999733652240642934342489, −15.32377659484724318820118364706, −15.13831557735662883329390065451, −13.64286432733788395014215193763, −13.15162022727472721856368525484, −12.58312846783355190101141196234, −11.92888519400616352808476147978, −11.09456696827511593979434184988, −10.19287614679174983879351331111, −9.61742283997970523427157217124, −8.669801871859741575384019283413, −8.027086681915384959848676515107, −7.62039411832284773143260445286, −6.6074373354946116239819986857, −5.756293947791475289523048419890, −5.306993892216174537158851413111, −4.914313199438528017251945416844, −4.00219502551289800372585941122, −2.68451549576100866854891310435, −1.45406196929848537820281045046, −0.70495422823118944919832265922, 0.44375384385173140305085851518, 1.258607346897596936328045914709, 2.50626150728548066736050143766, 3.08478514400157642184176247331, 4.141558878801909980980330276414, 4.55992118293303952522499575051, 5.31949466341380418860824452432, 6.472444212275936490464503772535, 7.1959919823893045391507891688, 7.6654363299938539730115547771, 8.74924248839712789209430090172, 9.75353856921839693247164796175, 10.232996078056763866381618386784, 10.76926260127101606283478686418, 11.37221010902066118331591659735, 11.840734776342894951258769086191, 12.59787668049302939515302944586, 13.68012582340132820504871926075, 13.86904793106284626926630732709, 14.845001227851310738494214910045, 15.88999662928384535370525250361, 16.328958800612896579512847384761, 17.16803002521799296657990181279, 17.83451519759754223263129853110, 18.31715735501494474551607963428

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