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Properties

Label 1-4003-4003.787-r0-0-0
Degree $1$
Conductor $4003$
Sign $-0.429 - 0.902i$
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  + (0.750 + 0.661i)2-s + (0.403 − 0.915i)3-s + (0.126 + 0.992i)4-s + (0.874 + 0.484i)5-s + (0.907 − 0.419i)6-s + (−0.232 + 0.972i)7-s + (−0.561 + 0.827i)8-s + (−0.674 − 0.738i)9-s + (0.336 + 0.941i)10-s + (0.267 − 0.963i)11-s + (0.958 + 0.284i)12-s + (−0.370 − 0.928i)13-s + (−0.817 + 0.576i)14-s + (0.796 − 0.605i)15-s + (−0.968 + 0.250i)16-s + (−0.161 − 0.986i)17-s + ⋯
L(s)  = 1  + (0.750 + 0.661i)2-s + (0.403 − 0.915i)3-s + (0.126 + 0.992i)4-s + (0.874 + 0.484i)5-s + (0.907 − 0.419i)6-s + (−0.232 + 0.972i)7-s + (−0.561 + 0.827i)8-s + (−0.674 − 0.738i)9-s + (0.336 + 0.941i)10-s + (0.267 − 0.963i)11-s + (0.958 + 0.284i)12-s + (−0.370 − 0.928i)13-s + (−0.817 + 0.576i)14-s + (0.796 − 0.605i)15-s + (−0.968 + 0.250i)16-s + (−0.161 − 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $-0.429 - 0.902i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ -0.429 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4559889139 - 0.7219683040i\)
\(L(\frac12)\) \(\approx\) \(0.4559889139 - 0.7219683040i\)
\(L(1)\) \(\approx\) \(1.430295195 + 0.2096250106i\)
\(L(1)\) \(\approx\) \(1.430295195 + 0.2096250106i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.750 + 0.661i)T \)
3 \( 1 + (0.403 - 0.915i)T \)
5 \( 1 + (0.874 + 0.484i)T \)
7 \( 1 + (-0.232 + 0.972i)T \)
11 \( 1 + (0.267 - 0.963i)T \)
13 \( 1 + (-0.370 - 0.928i)T \)
17 \( 1 + (-0.161 - 0.986i)T \)
19 \( 1 + (-0.773 + 0.633i)T \)
23 \( 1 + (-0.968 + 0.250i)T \)
29 \( 1 + (-0.773 - 0.633i)T \)
31 \( 1 + (-0.922 - 0.386i)T \)
37 \( 1 + (-0.773 - 0.633i)T \)
41 \( 1 + (0.267 + 0.963i)T \)
43 \( 1 + (0.590 - 0.806i)T \)
47 \( 1 + (-0.968 - 0.250i)T \)
53 \( 1 + (-0.968 - 0.250i)T \)
59 \( 1 + (-0.436 - 0.899i)T \)
61 \( 1 + (-0.0901 - 0.995i)T \)
67 \( 1 + (-0.817 - 0.576i)T \)
71 \( 1 + (0.403 - 0.915i)T \)
73 \( 1 + (0.468 + 0.883i)T \)
79 \( 1 + (-0.0901 + 0.995i)T \)
83 \( 1 + (0.750 + 0.661i)T \)
89 \( 1 + (-0.817 + 0.576i)T \)
97 \( 1 + (-0.817 - 0.576i)T \)
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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

−19.108846535033240199505222199220, −17.88235387419081940005956163275, −17.25848770954646173437668162533, −16.57052583223701853636437784658, −15.9836487174392251898680219486, −14.97985892995730436378609066461, −14.499338083626083545568918768216, −13.99394975508531441533940089165, −13.235756803704197999245391944222, −12.71349544824363053495203436367, −11.92349697552855783380457902632, −10.82288070507837592024636788074, −10.53167525690407883044316159860, −9.77002990273219013883679966935, −9.307042229805288695151166720713, −8.63505564885700852226141230776, −7.349432365582532232641083551309, −6.551107110019247995839602921, −5.822637885677675495376127782202, −4.85701166489506692383126943028, −4.37120817620408355902378766605, −3.91779248773785277095007999699, −2.91678160626752920121213997345, −1.90851136239865843441798467375, −1.61209611650828408810650152570, 0.13652943473757684053398798436, 1.863041913641410887619268067332, 2.35827173416898480831228984816, 3.14774856416523350658006041501, 3.68014711829465433583298566985, 5.17340277203573744195606044664, 5.73578300494675400930042232186, 6.19724102064015073878820294480, 6.81326681310937887050270922633, 7.77223761192941117793198002164, 8.22426294008357619080670417776, 9.14188764413389994245590297161, 9.62782611563910685289109310473, 10.98199383508013010641242970943, 11.56927469432028360467945487541, 12.537947756880340442260251821811, 12.76382154123010369884972714936, 13.6951969312312344640990872924, 14.04459658916948329403838799017, 14.76904186460226581454426144481, 15.27994191068263575766941957028, 16.161761244906936492298724048398, 16.94476119471706922607146687951, 17.624907062052454508440357369822, 18.25413768371240947961978659231

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