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Properties

Label 1-2003-2003.318-r1-0-0
Degree $1$
Conductor $2003$
Sign $0.521 - 0.853i$
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  − 2-s + (−0.222 + 0.974i)3-s + 4-s + (0.900 − 0.433i)5-s + (0.222 − 0.974i)6-s + (0.222 + 0.974i)7-s − 8-s + (−0.900 − 0.433i)9-s + (−0.900 + 0.433i)10-s − 11-s + (−0.222 + 0.974i)12-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.222 + 0.974i)15-s + 16-s + (−0.623 + 0.781i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.222 + 0.974i)3-s + 4-s + (0.900 − 0.433i)5-s + (0.222 − 0.974i)6-s + (0.222 + 0.974i)7-s − 8-s + (−0.900 − 0.433i)9-s + (−0.900 + 0.433i)10-s − 11-s + (−0.222 + 0.974i)12-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.222 + 0.974i)15-s + 16-s + (−0.623 + 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2003\)
Sign: $0.521 - 0.853i$
Analytic conductor: \(215.252\)
Root analytic conductor: \(215.252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2003} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2003,\ (1:\ ),\ 0.521 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04941658166 + 0.02769903170i\)
\(L(\frac12)\) \(\approx\) \(-0.04941658166 + 0.02769903170i\)
\(L(1)\) \(\approx\) \(0.5014291076 + 0.2814142991i\)
\(L(1)\) \(\approx\) \(0.5014291076 + 0.2814142991i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2003 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + (-0.900 - 0.433i)T \)
23 \( 1 - T \)
29 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (0.900 + 0.433i)T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (0.623 - 0.781i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−18.99049693561471947146614006606, −18.22255212697675064956437166251, −17.79465499347376735349384980568, −17.280992962151323469683802118019, −16.62898294153296309998426990456, −15.642757382982955057961542831083, −14.78314923072448278716399215767, −13.835500134470029198307596772186, −13.324871804760546193610723027574, −12.51806403697742752805429549554, −11.57889204248863085991852726869, −10.707673473480582824684172108340, −10.37111291667696315444127166865, −9.58067771174340109051040135348, −8.39212752240736362892262955277, −7.792333024751916216699088419392, −7.25399225716015846469109047017, −6.30971443042786292988731569618, −5.87744912197595464598279739830, −4.75475670520239899384760408420, −3.14406167325799480440013002795, −2.381945307969067941797639629779, −1.76927055075524760247064364585, −0.615503111042159928088313235976, −0.01964332089929131020058805192, 1.55048417302539493087655422904, 2.299880910413247516840890879472, 3.02835844381778251330593375815, 4.528400096281147464852147455073, 5.10873422002046000333994201218, 6.22333370248414670956495268334, 6.395725969147122494209048815529, 8.04730483031096828852659888796, 8.62845110056237200999766950033, 9.177212886585499240903354017753, 9.931138988895865024875782422995, 10.50592334864133597988008776383, 11.27251104216315725447235013863, 12.06666023907891845725386782157, 12.817162475931285766544553956998, 13.934378903805180343350317217052, 14.95045509618708541592705310291, 15.33359436534171051197364885174, 16.290966324278886961479358756007, 16.634844879257790647970640079517, 17.59132490368228348286500231567, 17.97301256489300334682951565018, 18.75777864134353438310368494556, 19.729372115749384373142287436640, 20.371950489460267992161000499311

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