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Properties

Label 1-4003-4003.10-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.0856 + 0.996i$
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  + (−0.674 + 0.738i)2-s + (0.997 + 0.0721i)3-s + (−0.0901 − 0.995i)4-s + (0.935 + 0.353i)5-s + (−0.725 + 0.687i)6-s + (0.874 − 0.484i)7-s + (0.796 + 0.605i)8-s + (0.989 + 0.143i)9-s + (−0.891 + 0.452i)10-s + (0.647 + 0.762i)11-s + (−0.0180 − 0.999i)12-s + (0.267 + 0.963i)13-s + (−0.232 + 0.972i)14-s + (0.907 + 0.419i)15-s + (−0.983 + 0.179i)16-s + (−0.994 − 0.108i)17-s + ⋯
L(s)  = 1  + (−0.674 + 0.738i)2-s + (0.997 + 0.0721i)3-s + (−0.0901 − 0.995i)4-s + (0.935 + 0.353i)5-s + (−0.725 + 0.687i)6-s + (0.874 − 0.484i)7-s + (0.796 + 0.605i)8-s + (0.989 + 0.143i)9-s + (−0.891 + 0.452i)10-s + (0.647 + 0.762i)11-s + (−0.0180 − 0.999i)12-s + (0.267 + 0.963i)13-s + (−0.232 + 0.972i)14-s + (0.907 + 0.419i)15-s + (−0.983 + 0.179i)16-s + (−0.994 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0856 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0856 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.0856 + 0.996i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.0856 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.033389240 + 1.866062459i\)
\(L(\frac12)\) \(\approx\) \(2.033389240 + 1.866062459i\)
\(L(1)\) \(\approx\) \(1.355218831 + 0.6413888077i\)
\(L(1)\) \(\approx\) \(1.355218831 + 0.6413888077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.674 + 0.738i)T \)
3 \( 1 + (0.997 + 0.0721i)T \)
5 \( 1 + (0.935 + 0.353i)T \)
7 \( 1 + (0.874 - 0.484i)T \)
11 \( 1 + (0.647 + 0.762i)T \)
13 \( 1 + (0.267 + 0.963i)T \)
17 \( 1 + (-0.994 - 0.108i)T \)
19 \( 1 + (0.590 - 0.806i)T \)
23 \( 1 + (-0.983 + 0.179i)T \)
29 \( 1 + (0.590 + 0.806i)T \)
31 \( 1 + (-0.817 + 0.576i)T \)
37 \( 1 + (0.590 + 0.806i)T \)
41 \( 1 + (0.647 - 0.762i)T \)
43 \( 1 + (-0.436 + 0.899i)T \)
47 \( 1 + (-0.983 - 0.179i)T \)
53 \( 1 + (-0.983 - 0.179i)T \)
59 \( 1 + (0.126 - 0.992i)T \)
61 \( 1 + (0.958 - 0.284i)T \)
67 \( 1 + (-0.232 - 0.972i)T \)
71 \( 1 + (0.997 + 0.0721i)T \)
73 \( 1 + (-0.947 - 0.319i)T \)
79 \( 1 + (0.958 + 0.284i)T \)
83 \( 1 + (-0.674 + 0.738i)T \)
89 \( 1 + (-0.232 + 0.972i)T \)
97 \( 1 + (-0.232 - 0.972i)T \)
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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

−18.25123801514437667752231957295, −17.91040451714937825736277306788, −17.26134284576070559777951082944, −16.301291011623930043087174545032, −15.77115881314335703184805894895, −14.678443210815455251329492602380, −14.17458869267668736217544254115, −13.39997712896662378317172932246, −12.95358117379908102576993809640, −12.12218532629007024544304326481, −11.39408246913929005477298611947, −10.582915890259190267294399833129, −9.88452943015120689945338735053, −9.27061715764492042329214746794, −8.599578591271704989748929975731, −8.19914120972151582068315815343, −7.52772443526267400552130219351, −6.35775324690833158046704293766, −5.63787954200667032189997289276, −4.50436732663324945270804235947, −3.8516357630014275299224700707, −2.90917534142523806093329023017, −2.22369333965967946989735855291, −1.6222442809328931814260338751, −0.86248611144077579611054518011, 1.277121540888474027638620289334, 1.76794146447993580090109419639, 2.38190938026769529153018859900, 3.67125236664636761164815829493, 4.65988944522254842534414972716, 4.99805489484944306626065787962, 6.39884654444641454908992351060, 6.80166934714109530401713472644, 7.409643056759772648371608516303, 8.267264780374077426906983752, 8.95583504571477651543171718383, 9.50912791630846301375369317375, 9.989102870451659456315839387786, 10.91502906719432340875184940700, 11.43633391254441715426888929908, 12.76563399546180442241146582096, 13.66592114636763647177774430175, 14.06815452239288527933814691647, 14.48666727816421793496138149065, 15.13637295131627473609965517879, 15.93512787553116329299407895236, 16.6016942530978949166812556108, 17.50272504646386255208983093143, 17.96708859823673130020265545242, 18.31793681991868802945664503686

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