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Properties

Label 1-4005-4005.203-r0-0-0
Degree $1$
Conductor $4005$
Sign $0.558 + 0.829i$
Analytic cond. $18.5991$
Root an. cond. $18.5991$
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  + (0.971 + 0.235i)2-s + (0.888 + 0.458i)4-s + (0.945 − 0.327i)7-s + (0.755 + 0.654i)8-s + (−0.981 + 0.189i)11-s + (−0.998 − 0.0475i)13-s + (0.995 − 0.0950i)14-s + (0.580 + 0.814i)16-s + (0.281 − 0.959i)17-s + (0.415 + 0.909i)19-s + (−0.998 − 0.0475i)22-s + (0.814 + 0.580i)23-s + (−0.959 − 0.281i)26-s + (0.989 + 0.142i)28-s + (0.327 + 0.945i)29-s + ⋯
L(s)  = 1  + (0.971 + 0.235i)2-s + (0.888 + 0.458i)4-s + (0.945 − 0.327i)7-s + (0.755 + 0.654i)8-s + (−0.981 + 0.189i)11-s + (−0.998 − 0.0475i)13-s + (0.995 − 0.0950i)14-s + (0.580 + 0.814i)16-s + (0.281 − 0.959i)17-s + (0.415 + 0.909i)19-s + (−0.998 − 0.0475i)22-s + (0.814 + 0.580i)23-s + (−0.959 − 0.281i)26-s + (0.989 + 0.142i)28-s + (0.327 + 0.945i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
Sign: $0.558 + 0.829i$
Analytic conductor: \(18.5991\)
Root analytic conductor: \(18.5991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4005} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4005,\ (0:\ ),\ 0.558 + 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.292033839 + 1.752963468i\)
\(L(\frac12)\) \(\approx\) \(3.292033839 + 1.752963468i\)
\(L(1)\) \(\approx\) \(2.015104380 + 0.4839603113i\)
\(L(1)\) \(\approx\) \(2.015104380 + 0.4839603113i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.971 + 0.235i)T \)
7 \( 1 + (0.945 - 0.327i)T \)
11 \( 1 + (-0.981 + 0.189i)T \)
13 \( 1 + (-0.998 - 0.0475i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (0.814 + 0.580i)T \)
29 \( 1 + (0.327 + 0.945i)T \)
31 \( 1 + (0.995 - 0.0950i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.888 - 0.458i)T \)
43 \( 1 + (-0.189 - 0.981i)T \)
47 \( 1 + (-0.998 + 0.0475i)T \)
53 \( 1 + (0.540 - 0.841i)T \)
59 \( 1 + (0.888 + 0.458i)T \)
61 \( 1 + (0.786 + 0.618i)T \)
67 \( 1 + (-0.998 - 0.0475i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.580 - 0.814i)T \)
83 \( 1 + (0.690 + 0.723i)T \)
97 \( 1 + (-0.945 + 0.327i)T \)
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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

−18.4503153100969446970047927831, −17.6578892847658667481247480744, −16.986246193870999772887749861420, −16.17495099243035057254906358011, −15.294273624559894893673185405844, −15.02973319380032001871871461723, −14.295433772259950366412761054929, −13.57579454292299825153141488979, −12.90219281288497157382725998763, −12.27630823822136049777899125980, −11.55490610106131298753132824491, −10.98354153803232593935973377099, −10.28489509327799540597950271282, −9.5839458047679294417847236486, −8.40751876589475559863356924932, −7.85710989226437888508134817944, −7.06510835028589327175726708695, −6.24563014422082966421252956834, −5.37892052904257676653624999302, −4.87581856794502485205022000667, −4.32793216861850360279151764937, −3.14127310652284190654964848409, −2.54891423476981496316339780186, −1.85684512324813940231356696447, −0.76310476595267169448352877698, 1.09677082544018113981027500419, 2.059744138342748084131520979228, 2.81708898381274730033303268822, 3.55330246645784995468588345943, 4.58475012239364283994741835296, 5.20611387168510381247749051179, 5.40988756685705863887443120621, 6.812745804285735938730855480767, 7.235113590502270812258255679808, 7.94579627064400056096861916056, 8.52745773741395925001594369871, 9.890027044224821320115444144979, 10.33526114480878447004136443555, 11.24521438299963031996051396769, 11.88419503527028044964276664014, 12.34616712581901610262131698829, 13.3849434288103549449413029483, 13.71317378009108451716629142351, 14.6165774514150967400421701131, 14.95142630636458227362745922021, 15.7742534048076147150574843617, 16.44392103029919255524776388356, 17.12829505399883403511691957903, 17.76115126764999391886205927539, 18.52168864923017096800576301562

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