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Properties

Label 1-4235-4235.48-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.994 - 0.109i$
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  + (−0.856 − 0.516i)2-s + (−0.587 + 0.809i)3-s + (0.466 + 0.884i)4-s + (0.921 − 0.389i)6-s + (0.0570 − 0.998i)8-s + (−0.309 − 0.951i)9-s + (−0.989 − 0.142i)12-s + (0.170 + 0.985i)13-s + (−0.564 + 0.825i)16-s + (0.113 + 0.993i)17-s + (−0.226 + 0.974i)18-s + (−0.870 + 0.491i)19-s + (0.281 − 0.959i)23-s + (0.774 + 0.633i)24-s + (0.362 − 0.931i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.856 − 0.516i)2-s + (−0.587 + 0.809i)3-s + (0.466 + 0.884i)4-s + (0.921 − 0.389i)6-s + (0.0570 − 0.998i)8-s + (−0.309 − 0.951i)9-s + (−0.989 − 0.142i)12-s + (0.170 + 0.985i)13-s + (−0.564 + 0.825i)16-s + (0.113 + 0.993i)17-s + (−0.226 + 0.974i)18-s + (−0.870 + 0.491i)19-s + (0.281 − 0.959i)23-s + (0.774 + 0.633i)24-s + (0.362 − 0.931i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.994 - 0.109i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.994 - 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6921229807 - 0.03796538520i\)
\(L(\frac12)\) \(\approx\) \(0.6921229807 - 0.03796538520i\)
\(L(1)\) \(\approx\) \(0.5741411041 + 0.04034959098i\)
\(L(1)\) \(\approx\) \(0.5741411041 + 0.04034959098i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.856 - 0.516i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (0.170 + 0.985i)T \)
17 \( 1 + (0.113 + 0.993i)T \)
19 \( 1 + (-0.870 + 0.491i)T \)
23 \( 1 + (0.281 - 0.959i)T \)
29 \( 1 + (-0.198 - 0.980i)T \)
31 \( 1 + (-0.897 + 0.441i)T \)
37 \( 1 + (0.717 - 0.696i)T \)
41 \( 1 + (-0.0855 - 0.996i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (0.226 + 0.974i)T \)
53 \( 1 + (0.825 - 0.564i)T \)
59 \( 1 + (0.0855 - 0.996i)T \)
61 \( 1 + (-0.516 - 0.856i)T \)
67 \( 1 + (-0.755 + 0.654i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (-0.336 - 0.941i)T \)
79 \( 1 + (0.254 - 0.967i)T \)
83 \( 1 + (0.999 + 0.0285i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.633 + 0.774i)T \)
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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

−18.30074759286908974375876485301, −17.82384878162196812366730490672, −17.01089130533505726848476522310, −16.652812701729295954420773320439, −15.81711420179268593105463963598, −15.145532666665544045700314678153, −14.47410523878058262871982403685, −13.46063190049738811078505802641, −13.121359980942640493621425054505, −12.05912394670363461596395265344, −11.46804965402561321505428978396, −10.77291347355869022818938671610, −10.24738469831811411420358591161, −9.25137233399167240620830656393, −8.6384626111159369355896163487, −7.7942802821526847817822635123, −7.27611636279418747081982253217, −6.7184996925548921345014988291, −5.73566475984260278648795318785, −5.44538200149994850654914159310, −4.51323486022340426908791079299, −3.06452574731851521771459797355, −2.31875006315834421092776819218, −1.36446112938399118562172660677, −0.64497277974779929289672520962, 0.454340176641744920115958272535, 1.61571329719797224073103584847, 2.36256486143517666801245177653, 3.47532566249708581646815034912, 4.07231013730117025786845464442, 4.66125898228663341590933413165, 5.99169731558425735443234267083, 6.341412091386279010875303216069, 7.29114093073787443871443694043, 8.20660805951206690834683575769, 8.90980482163584736457594460673, 9.41520449742333144097424149644, 10.20803550367649845101370858129, 10.860810536717152701030217885885, 11.16092803394465382810832518371, 12.20990052602784625396034538594, 12.47690771312953020704948294508, 13.44313882113352286276253246294, 14.628091827936036522745662094113, 14.978742624639333112974783487714, 16.164455373604063343315688448729, 16.29213184361971672328523047725, 17.10759569872934118847356699149, 17.54071391549751581611556476458, 18.34108117575847535526343328323

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