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Properties

Label 1-4005-4005.3559-r0-0-0
Degree $1$
Conductor $4005$
Sign $0.173 - 0.984i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − 19-s + (0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 26-s + 28-s + (0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − 19-s + (0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 26-s + 28-s + (0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(18.5991\)
Root analytic conductor: \(18.5991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4005} (3559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4005,\ (0:\ ),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8878001504 - 0.7449527787i\)
\(L(\frac12)\) \(\approx\) \(0.8878001504 - 0.7449527787i\)
\(L(1)\) \(\approx\) \(0.8849157470 - 0.4046923851i\)
\(L(1)\) \(\approx\) \(0.8849157470 - 0.4046923851i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−18.66963132900790425692002936483, −17.554193509992608747496467793308, −17.24737204381526670072384550521, −16.453777659991771433609223645803, −15.960600862459277932098135673193, −15.36105724018266050578671228844, −14.38652345392661763800424557791, −13.95981396443172392472825704892, −13.22589138281502617267010303915, −12.84633566247067801707669243241, −11.852148082460254048714885467458, −11.1390990810693241166901759453, −10.35182754083715838158754659786, −9.417521793936754270634637581070, −8.82993669932842404190903553966, −7.97703946043439510309254926684, −7.360437181286818672230468768059, −6.55290682331309406060685324439, −6.16121811332650346049102775420, −5.17113407340499261068857701295, −4.346522277393404328917570905470, −3.87762946385863756296243241773, −2.960400951454234629956740639740, −2.091498376052380275881256119993, −0.56473569620525145028335043035, 0.455105641139777981319028705421, 1.81895197729198270647960135, 2.5889373406763059453120229989, 2.84363544053509658466959645630, 4.16972706943197338501881087223, 4.63915390438944422737072520864, 5.45219469742989536983091379796, 6.23773615590183465976083492021, 6.819701638397192059646790806497, 8.14823828426824834369232950785, 8.61245349170411162249001728183, 9.694163705120440199969813877180, 9.97698515356324577003072989287, 10.7781507596654624915737192410, 11.54392142652299118443488505124, 12.3320466969375224744664905155, 12.809497036642296565700156811343, 13.1814500118095886127559447634, 14.24488164452257155689248086908, 14.967136206327777126560825369850, 15.37940138547074834370807705161, 16.02380749815336461959558698039, 17.192939961188615013971808313488, 17.95198768085237582349673785739, 18.29893729382211777809200029114

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