Properties

Label 1-448-448.67-r1-0-0
Degree $1$
Conductor $448$
Sign $-0.410 - 0.911i$
Analytic cond. $48.1442$
Root an. cond. $48.1442$
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  + (−0.130 − 0.991i)3-s + (−0.991 − 0.130i)5-s + (−0.965 + 0.258i)9-s + (0.793 + 0.608i)11-s + (0.382 − 0.923i)13-s + i·15-s + (0.866 − 0.5i)17-s + (0.608 + 0.793i)19-s + (−0.965 + 0.258i)23-s + (0.965 + 0.258i)25-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.991 + 0.130i)37-s + ⋯
L(s)  = 1  + (−0.130 − 0.991i)3-s + (−0.991 − 0.130i)5-s + (−0.965 + 0.258i)9-s + (0.793 + 0.608i)11-s + (0.382 − 0.923i)13-s + i·15-s + (0.866 − 0.5i)17-s + (0.608 + 0.793i)19-s + (−0.965 + 0.258i)23-s + (0.965 + 0.258i)25-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.991 + 0.130i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.410 - 0.911i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ -0.410 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7555022804 - 1.169083999i\)
\(L(\frac12)\) \(\approx\) \(0.7555022804 - 1.169083999i\)
\(L(1)\) \(\approx\) \(0.8451388128 - 0.3708319551i\)
\(L(1)\) \(\approx\) \(0.8451388128 - 0.3708319551i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.130 - 0.991i)T \)
5 \( 1 + (-0.991 - 0.130i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.608 + 0.793i)T \)
23 \( 1 + (-0.965 + 0.258i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.991 + 0.130i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.793 - 0.608i)T \)
59 \( 1 + (-0.608 + 0.793i)T \)
61 \( 1 + (0.793 - 0.608i)T \)
67 \( 1 + (-0.130 - 0.991i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.258 - 0.965i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (-0.258 - 0.965i)T \)
97 \( 1 - T \)
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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

−23.71084668603166717920938033663, −23.35167928118492805386208333668, −22.10147329442788107369110474138, −21.75418042081640634862950230246, −20.64168202069336660204592067510, −19.781711130524703006632313934469, −19.1495461366035198223250934410, −18.01696326258964334715537179472, −16.810743243072137491513690243146, −16.24049466339810877442633054868, −15.5423999512015633604784412578, −14.50172379597400650234124372068, −13.94555289573938826340135853077, −12.29899606424076077624728831294, −11.568410795837052252857990792088, −10.9367132673515339647370356189, −9.82743012140710018729200865486, −8.87250319745532005085220533314, −8.10358270031592653609487031010, −6.78672233091424445067909249665, −5.794958909994371397481704968927, −4.49499641541307758851054896721, −3.85150618095376220576841788904, −2.920058048903496193860043747184, −0.97901400085916382742207504404, 0.495311982385449011909484648679, 1.48895096333752660489615429114, 2.975400560921401439363874275573, 3.97827914134647879615651145422, 5.33813561001920435202751993667, 6.31161371388696433537175602279, 7.53855043778234026245113295864, 7.85031555878818321460105464175, 9.01667936397016579102220544842, 10.27697544791707308191978061998, 11.48395010643514594095536119946, 12.11113642251228445670321669683, 12.71868102972059834556369367742, 13.931009664571324105137624749832, 14.674888875402194834181883458002, 15.77370344282105981842730313068, 16.63613135281312828357058878944, 17.612933722886663592836023498901, 18.44347582140138290166353141043, 19.16354646650213257254707660481, 20.16976653611148762970788384773, 20.45859866078055904961755983313, 22.21006200420350485624217463609, 22.825431696407192194985701036364, 23.50898295910112529447458032402

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