Properties

Label 1-2003-2003.45-r1-0-0
Degree $1$
Conductor $2003$
Sign $-0.256 + 0.966i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.354 − 0.935i)2-s + (0.120 − 0.992i)3-s + (−0.748 − 0.663i)4-s + (−0.568 + 0.822i)5-s + (−0.885 − 0.464i)6-s + (−0.885 − 0.464i)7-s + (−0.885 + 0.464i)8-s + (−0.970 − 0.239i)9-s + (0.568 + 0.822i)10-s + (0.970 − 0.239i)11-s + (−0.748 + 0.663i)12-s + (0.568 + 0.822i)13-s + (−0.748 + 0.663i)14-s + (0.748 + 0.663i)15-s + (0.120 + 0.992i)16-s + (0.354 + 0.935i)17-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)2-s + (0.120 − 0.992i)3-s + (−0.748 − 0.663i)4-s + (−0.568 + 0.822i)5-s + (−0.885 − 0.464i)6-s + (−0.885 − 0.464i)7-s + (−0.885 + 0.464i)8-s + (−0.970 − 0.239i)9-s + (0.568 + 0.822i)10-s + (0.970 − 0.239i)11-s + (−0.748 + 0.663i)12-s + (0.568 + 0.822i)13-s + (−0.748 + 0.663i)14-s + (0.748 + 0.663i)15-s + (0.120 + 0.992i)16-s + (0.354 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2003\)
Sign: $-0.256 + 0.966i$
Analytic conductor: \(215.252\)
Root analytic conductor: \(215.252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2003} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2003,\ (1:\ ),\ -0.256 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2682902957 - 0.3487186123i\)
\(L(\frac12)\) \(\approx\) \(-0.2682902957 - 0.3487186123i\)
\(L(1)\) \(\approx\) \(0.6949282346 - 0.6002611512i\)
\(L(1)\) \(\approx\) \(0.6949282346 - 0.6002611512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2003 \( 1 \)
good2 \( 1 + (0.354 - 0.935i)T \)
3 \( 1 + (0.120 - 0.992i)T \)
5 \( 1 + (-0.568 + 0.822i)T \)
7 \( 1 + (-0.885 - 0.464i)T \)
11 \( 1 + (0.970 - 0.239i)T \)
13 \( 1 + (0.568 + 0.822i)T \)
17 \( 1 + (0.354 + 0.935i)T \)
19 \( 1 + (0.120 + 0.992i)T \)
23 \( 1 + (0.970 - 0.239i)T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (0.748 - 0.663i)T \)
37 \( 1 + (-0.120 - 0.992i)T \)
41 \( 1 + (-0.885 + 0.464i)T \)
43 \( 1 + (-0.885 + 0.464i)T \)
47 \( 1 + (0.885 - 0.464i)T \)
53 \( 1 + (0.885 - 0.464i)T \)
59 \( 1 + (0.120 - 0.992i)T \)
61 \( 1 + (-0.885 + 0.464i)T \)
67 \( 1 + (-0.568 - 0.822i)T \)
71 \( 1 + (-0.885 + 0.464i)T \)
73 \( 1 + (0.120 + 0.992i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.120 + 0.992i)T \)
89 \( 1 + (-0.748 - 0.663i)T \)
97 \( 1 + (-0.568 - 0.822i)T \)
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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

−20.39554763982829773422106586812, −19.688346645751994176679378879827, −18.85442815299480214027910501972, −17.824282809780410199126417543578, −16.87079618449497024799182670338, −16.64645767909173290535691504931, −15.65899643615973715053053059840, −15.45894063919817888219566565188, −14.790795462253198388719068033725, −13.61413080750846051300641417106, −13.2359821707165698897429312542, −12.08485783421930410827838551735, −11.76634108301324653954540863630, −10.49074182530494698054464821826, −9.40370442961020206512232426557, −9.050330786577026612022207922151, −8.50506180128813854477583812097, −7.42740389223625417919260144720, −6.62923258502278211650689534663, −5.58955078234085084385944454427, −5.07957287876826423088255566641, −4.30440376211881667725621574527, −3.38835857417501918198958937214, −2.98605837907726743005554002766, −0.92430454680001367604248549826, 0.08724257474811117871554901662, 1.101609337343425760832733698757, 1.90133433987321618638738900075, 2.95472829713015613019399893366, 3.685415868585833450775415430662, 4.078750783569361285321666633996, 5.73814118647408695780912112601, 6.37217052930903497830139931346, 6.92697367486645467972338523145, 7.99548335857024023323673643829, 8.81070497135337938309282784138, 9.66201185196617682119742266137, 10.532097084609442785205451034217, 11.307603612854183100790211622359, 11.85515790817573374042122916948, 12.54825293544903615249629959721, 13.344945929476168170109906317211, 13.93197533188084257973930450643, 14.608705438715729342400169617759, 15.221267957516441064034043391415, 16.61258279890011429432880066234, 17.09577810748014489611279094114, 18.279375645787379138666689006173, 18.824336372914340634040293908920, 19.24012458279731684213942238720

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