Properties

Label 1-4235-4235.1493-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.503 + 0.864i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.703 − 0.710i)2-s + (0.994 − 0.104i)3-s + (−0.00951 + 0.999i)4-s + (−0.774 − 0.633i)6-s + (0.717 − 0.696i)8-s + (0.978 − 0.207i)9-s + (0.0950 + 0.995i)12-s + (−0.676 + 0.736i)13-s + (−0.999 − 0.0190i)16-s + (−0.475 + 0.879i)17-s + (−0.836 − 0.548i)18-s + (−0.991 − 0.132i)19-s + (−0.945 − 0.327i)23-s + (0.640 − 0.768i)24-s + (0.999 − 0.0380i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.703 − 0.710i)2-s + (0.994 − 0.104i)3-s + (−0.00951 + 0.999i)4-s + (−0.774 − 0.633i)6-s + (0.717 − 0.696i)8-s + (0.978 − 0.207i)9-s + (0.0950 + 0.995i)12-s + (−0.676 + 0.736i)13-s + (−0.999 − 0.0190i)16-s + (−0.475 + 0.879i)17-s + (−0.836 − 0.548i)18-s + (−0.991 − 0.132i)19-s + (−0.945 − 0.327i)23-s + (0.640 − 0.768i)24-s + (0.999 − 0.0380i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.503 + 0.864i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.503 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2141935797 + 0.3727025549i\)
\(L(\frac12)\) \(\approx\) \(0.2141935797 + 0.3727025549i\)
\(L(1)\) \(\approx\) \(0.8309085909 - 0.1450332959i\)
\(L(1)\) \(\approx\) \(0.8309085909 - 0.1450332959i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.703 - 0.710i)T \)
3 \( 1 + (0.994 - 0.104i)T \)
13 \( 1 + (-0.676 + 0.736i)T \)
17 \( 1 + (-0.475 + 0.879i)T \)
19 \( 1 + (-0.991 - 0.132i)T \)
23 \( 1 + (-0.945 - 0.327i)T \)
29 \( 1 + (0.941 + 0.336i)T \)
31 \( 1 + (-0.398 + 0.917i)T \)
37 \( 1 + (0.318 + 0.948i)T \)
41 \( 1 + (0.362 - 0.931i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (0.836 - 0.548i)T \)
53 \( 1 + (-0.0190 - 0.999i)T \)
59 \( 1 + (-0.988 - 0.151i)T \)
61 \( 1 + (-0.964 + 0.263i)T \)
67 \( 1 + (0.998 - 0.0475i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (-0.424 + 0.905i)T \)
79 \( 1 + (-0.0665 - 0.997i)T \)
83 \( 1 + (-0.389 + 0.921i)T \)
89 \( 1 + (-0.235 - 0.971i)T \)
97 \( 1 + (-0.170 + 0.985i)T \)
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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

−18.18208600063369777320220493930, −17.51440989485198919436902901049, −16.763645195422227772462228870100, −16.00054766732870857274565999769, −15.434592755841210022406212711320, −14.91813411502425951867081872295, −14.22655734864949543312072630421, −13.657644370321721563414368844782, −12.91206958412555085290520955151, −12.042643036739774592734822113, −10.98112737841163588954227451083, −10.346880193953634691272672990808, −9.626925870600479590424192529539, −9.202819448668112749419022861583, −8.31420068863601057849945630253, −7.82975518844811156849766942629, −7.24108266485767613467093087200, −6.40721742137486779885577248351, −5.62146924158798375048137371445, −4.640814125778297549183811467122, −4.11827564547693911485044228264, −2.80620910838423677757465069787, −2.295999946087840228134104000229, −1.33376328269715936312194926383, −0.12192804551579147666278180914, 1.34819081695117171938040566908, 2.01692334534643759457362657489, 2.58551711940785357483371929467, 3.50341218719496330516163989539, 4.19314563536854657990866568969, 4.79742504346475204435243570783, 6.38893081279822070580357997769, 6.89862995612846167648790771618, 7.727992135969265620298672817656, 8.51622765073141360768475334236, 8.77543611308542120398847051927, 9.65947138160142574274487724939, 10.28668458404027002971778522119, 10.81815558091584192304039958053, 11.91580257854148357810436914615, 12.39248981839990657708307164882, 13.05479863997749681983170238416, 13.79502488955165051093970150115, 14.43879434115929566599387296637, 15.22515192384388020312810596194, 15.94403337762896054181327248568, 16.71308359913235182790331786932, 17.37036521030940070039675405438, 18.11471258454673303027275085429, 18.74277564529905546915062188918

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