↑

Properties

Label 1-2003-2003.234-r1-0-0
Degree $1$
Conductor $2003$
Sign $0.258 + 0.966i$
Analytic cond. $215.252$
Root an. cond. $215.252$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.142 − 0.989i)2-s + (0.685 − 0.728i)3-s + (−0.959 − 0.281i)4-s + (0.0611 − 0.998i)5-s + (−0.623 − 0.781i)6-s + (−0.947 + 0.320i)7-s + (−0.415 + 0.909i)8-s + (−0.0611 − 0.998i)9-s + (−0.979 − 0.202i)10-s + (0.142 + 0.989i)11-s + (−0.862 + 0.505i)12-s + (0.947 + 0.320i)13-s + (0.182 + 0.983i)14-s + (−0.685 − 0.728i)15-s + (0.841 + 0.540i)16-s + (0.900 − 0.433i)17-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (0.685 − 0.728i)3-s + (−0.959 − 0.281i)4-s + (0.0611 − 0.998i)5-s + (−0.623 − 0.781i)6-s + (−0.947 + 0.320i)7-s + (−0.415 + 0.909i)8-s + (−0.0611 − 0.998i)9-s + (−0.979 − 0.202i)10-s + (0.142 + 0.989i)11-s + (−0.862 + 0.505i)12-s + (0.947 + 0.320i)13-s + (0.182 + 0.983i)14-s + (−0.685 − 0.728i)15-s + (0.841 + 0.540i)16-s + (0.900 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2003\)
Sign: $0.258 + 0.966i$
Analytic conductor: \(215.252\)
Root analytic conductor: \(215.252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2003} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2003,\ (1:\ ),\ 0.258 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.173652113 - 0.9013358378i\)
\(L(\frac12)\) \(\approx\) \(-1.173652113 - 0.9013358378i\)
\(L(1)\) \(\approx\) \(0.5663264202 - 0.9778919674i\)
\(L(1)\) \(\approx\) \(0.5663264202 - 0.9778919674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2003 \( 1 \)
good2 \( 1 + (0.142 - 0.989i)T \)
3 \( 1 + (0.685 - 0.728i)T \)
5 \( 1 + (0.0611 - 0.998i)T \)
7 \( 1 + (-0.947 + 0.320i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (0.947 + 0.320i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 + (-0.0611 - 0.998i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (-0.685 - 0.728i)T \)
31 \( 1 + (-0.794 - 0.607i)T \)
37 \( 1 + (0.714 - 0.699i)T \)
41 \( 1 + (-0.0203 - 0.999i)T \)
43 \( 1 + (-0.986 + 0.162i)T \)
47 \( 1 + (0.970 + 0.242i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (0.488 + 0.872i)T \)
61 \( 1 + (-0.917 + 0.396i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (0.0203 - 0.999i)T \)
79 \( 1 + (0.986 - 0.162i)T \)
83 \( 1 + (-0.970 - 0.242i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.742 + 0.670i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.30981894487803233520969265287, −19.47587933469182606566001105317, −18.60448251439773577943832680216, −18.51174197287265814651452502881, −17.03014526337043264105569931633, −16.56733641203666177315775545241, −15.90459854069259638782352754187, −15.24666037243678477320117948882, −14.56775790848030156476130840834, −13.89520142578746463741715363621, −13.44226889142973628891971029751, −12.572948635188302810238499440891, −11.21499570005874904413430696823, −10.47505627704068949656544440242, −9.8007485730012357473186760023, −9.1309021816012167983867071824, −8.17834591869540678346091476148, −7.667424988359775708961965179001, −6.66920851982958939634815966211, −5.94005092259584313772188944638, −5.34902818243207836416724215071, −3.82104796911407766709988407576, −3.54796042584160101871998831421, −3.029562453077519584995979774996, −1.32915835142771688325617303298, 0.25505170205368003396294296652, 0.957244919791327512410863057093, 1.97015010987419613523510244574, 2.57254030455916945954431796890, 3.65118314806580091780002681603, 4.22981858681383424569145461406, 5.354670219411245064016114283567, 6.15279028861103806517844682137, 7.17027523747351694469329608491, 8.1446458925521363987777187041, 9.01884511482764347171188519359, 9.31542517704281266947886999986, 10.02516419917219647779296687820, 11.25969575975593547973350336802, 12.12384050161966085747106875222, 12.5440539456866400544375133248, 13.22077938772537089102499717425, 13.61449328550475656803419989432, 14.633911318441320167059749559118, 15.35255704367011489661730363013, 16.3577884355080500540619118311, 17.18571908786173853164651408705, 18.07262681222901862312146012900, 18.65649556741898563566770403857, 19.30293121032688579515538740542

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