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Properties

Label 1-4100-4100.3719-r0-0-0
Degree $1$
Conductor $4100$
Sign $0.464 + 0.885i$
Analytic cond. $19.0403$
Root an. cond. $19.0403$
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  + (−0.891 + 0.453i)3-s + (0.453 − 0.891i)7-s + (0.587 − 0.809i)9-s + (−0.891 + 0.453i)11-s + (−0.987 − 0.156i)13-s + (−0.987 + 0.156i)17-s + (−0.156 + 0.987i)19-s + i·21-s + (−0.309 − 0.951i)23-s + (−0.156 + 0.987i)27-s + (0.987 + 0.156i)29-s + (−0.809 − 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.309 − 0.951i)37-s + (0.951 − 0.309i)39-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)3-s + (0.453 − 0.891i)7-s + (0.587 − 0.809i)9-s + (−0.891 + 0.453i)11-s + (−0.987 − 0.156i)13-s + (−0.987 + 0.156i)17-s + (−0.156 + 0.987i)19-s + i·21-s + (−0.309 − 0.951i)23-s + (−0.156 + 0.987i)27-s + (0.987 + 0.156i)29-s + (−0.809 − 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.309 − 0.951i)37-s + (0.951 − 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4100\)    =    \(2^{2} \cdot 5^{2} \cdot 41\)
Sign: $0.464 + 0.885i$
Analytic conductor: \(19.0403\)
Root analytic conductor: \(19.0403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4100} (3719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4100,\ (0:\ ),\ 0.464 + 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5253862338 + 0.3174954594i\)
\(L(\frac12)\) \(\approx\) \(0.5253862338 + 0.3174954594i\)
\(L(1)\) \(\approx\) \(0.6628670556 + 0.03678492142i\)
\(L(1)\) \(\approx\) \(0.6628670556 + 0.03678492142i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.891 + 0.453i)T \)
7 \( 1 + (0.453 - 0.891i)T \)
11 \( 1 + (-0.891 + 0.453i)T \)
13 \( 1 + (-0.987 - 0.156i)T \)
17 \( 1 + (-0.987 + 0.156i)T \)
19 \( 1 + (-0.156 + 0.987i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (0.453 + 0.891i)T \)
71 \( 1 + (-0.453 + 0.891i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.891 + 0.453i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.156 + 0.987i)T \)
97 \( 1 + (0.156 - 0.987i)T \)
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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

−18.184632262344304388356651621163, −17.69848923057787122112290050730, −17.176787466988774775183917089678, −16.25213618831123728484229732939, −15.615810924225510020785900025446, −15.17706178795139233349943289065, −14.12284730414321570469178145075, −13.368416072133325667803580972421, −12.87600257537597527014676579892, −11.91096716730882046450743797835, −11.68440467910535543961627338250, −10.84127313650670359870510016463, −10.244464643741772519425709355787, −9.277613377034384222271017174956, −8.54528542932637078835006229070, −7.777267425669346250262648130733, −7.058717876901828145295543209623, −6.37588756277489890532213557660, −5.48399525020246833166404691707, −5.033981834393688991434019310359, −4.42150886013119049576154571682, −2.98289338840795948610355142977, −2.324748889682259724178438495183, −1.579270221462572737206461425949, −0.298008126621382907780115931622, 0.622762586725091197249704063133, 1.80514540886695468313153076210, 2.62189802083732834440923850052, 3.94202795809477406634186593565, 4.31402212348901637326865737557, 5.11428776015427896781525857001, 5.69399714976563235619606693387, 6.75045654711162790303156709701, 7.21644827263436252071541941847, 8.033396856860215933051312714597, 8.8819334458155530888233289394, 9.98844239816678629880572580011, 10.29356929868343634215830387537, 10.83337608186890209589816850787, 11.65622928225905801587918951386, 12.40843120067197916441119229997, 12.90018794855289356805422338548, 13.80360408960573441783036712020, 14.67105489132701995943748543258, 15.125574958176891223985597660723, 16.00776854181265158085203907587, 16.568527754324623601179685551233, 17.18551244617362743954266779918, 17.82853447526679088132547172650, 18.24049323933703927913406670431

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