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Properties

Label 1-4003-4003.785-r0-0-0
Degree $1$
Conductor $4003$
Sign $-0.267 - 0.963i$
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  + (0.197 − 0.980i)2-s + (0.590 + 0.806i)3-s + (−0.922 − 0.386i)4-s + (−0.0180 − 0.999i)5-s + (0.907 − 0.419i)6-s + (0.958 − 0.284i)7-s + (−0.561 + 0.827i)8-s + (−0.302 + 0.953i)9-s + (−0.983 − 0.179i)10-s + (0.267 − 0.963i)11-s + (−0.232 − 0.972i)12-s + (−0.370 − 0.928i)13-s + (−0.0901 − 0.995i)14-s + (0.796 − 0.605i)15-s + (0.700 + 0.713i)16-s + (−0.161 − 0.986i)17-s + ⋯
L(s)  = 1  + (0.197 − 0.980i)2-s + (0.590 + 0.806i)3-s + (−0.922 − 0.386i)4-s + (−0.0180 − 0.999i)5-s + (0.907 − 0.419i)6-s + (0.958 − 0.284i)7-s + (−0.561 + 0.827i)8-s + (−0.302 + 0.953i)9-s + (−0.983 − 0.179i)10-s + (0.267 − 0.963i)11-s + (−0.232 − 0.972i)12-s + (−0.370 − 0.928i)13-s + (−0.0901 − 0.995i)14-s + (0.796 − 0.605i)15-s + (0.700 + 0.713i)16-s + (−0.161 − 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $-0.267 - 0.963i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ -0.267 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.589523139 - 2.091331060i\)
\(L(\frac12)\) \(\approx\) \(1.589523139 - 2.091331060i\)
\(L(1)\) \(\approx\) \(1.284200210 - 0.7844852125i\)
\(L(1)\) \(\approx\) \(1.284200210 - 0.7844852125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.197 - 0.980i)T \)
3 \( 1 + (0.590 + 0.806i)T \)
5 \( 1 + (-0.0180 - 0.999i)T \)
7 \( 1 + (0.958 - 0.284i)T \)
11 \( 1 + (0.267 - 0.963i)T \)
13 \( 1 + (-0.370 - 0.928i)T \)
17 \( 1 + (-0.161 - 0.986i)T \)
19 \( 1 + (0.935 + 0.353i)T \)
23 \( 1 + (0.700 + 0.713i)T \)
29 \( 1 + (0.935 - 0.353i)T \)
31 \( 1 + (0.126 + 0.992i)T \)
37 \( 1 + (0.935 - 0.353i)T \)
41 \( 1 + (0.267 + 0.963i)T \)
43 \( 1 + (0.403 + 0.915i)T \)
47 \( 1 + (0.700 - 0.713i)T \)
53 \( 1 + (0.700 - 0.713i)T \)
59 \( 1 + (0.997 + 0.0721i)T \)
61 \( 1 + (-0.817 + 0.576i)T \)
67 \( 1 + (-0.0901 + 0.995i)T \)
71 \( 1 + (0.590 + 0.806i)T \)
73 \( 1 + (0.468 + 0.883i)T \)
79 \( 1 + (-0.817 - 0.576i)T \)
83 \( 1 + (0.197 - 0.980i)T \)
89 \( 1 + (-0.0901 - 0.995i)T \)
97 \( 1 + (-0.0901 + 0.995i)T \)
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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

−18.461333643828097403445805491163, −18.06239544828943978140003438650, −17.34202182836019335491983940273, −16.92267188150465042022775070319, −15.5390507535379525322844383650, −15.15431159463430644845689280531, −14.58584336582128910664620394568, −14.084640020437618322370669722280, −13.59128397995084758163538182189, −12.51529637944287062851347151395, −12.09713083919406210264693356751, −11.28003988416664115690588076405, −10.27497341475446312675315582121, −9.29366134421461671800303442975, −8.889035457055721510707011437445, −7.81659033814039423359640632755, −7.57273277530778398570464931163, −6.73965711274002508161666943219, −6.36515708322733867537469106987, −5.36246211858344486882601567458, −4.414447223216848784784727156268, −3.82823851622953188254363753641, −2.681564305474191054733516031737, −2.128716456457685861456947148307, −1.03889183740616015889183430388, 0.79474974465596479509006511276, 1.314344062490691222214748308411, 2.54609229551700051770953889272, 3.11351248589837501617061710057, 3.9852842386720486849662233660, 4.64740166821375842817840016878, 5.26890842058824996186247267868, 5.678213944249672918961528958814, 7.43767998987678519938391003087, 8.1847548048762780098168018560, 8.64439028177357725531035218367, 9.367727959707622294733678810469, 9.96778699960093772604825475672, 10.688264900713345889432440887427, 11.58002276759117421580071370614, 11.7421839726098381136000161468, 12.98773817521960027626033995192, 13.45045640925356169009497394203, 14.16243017775286589449513204282, 14.58946346212994988519305017840, 15.56662930022643419701751860562, 16.18695997066249700825846803479, 16.97027262330551136930711019782, 17.6798719592781446928092722970, 18.288441474081800738050106613060

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