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Properties

Label 1-448-448.51-r1-0-0
Degree $1$
Conductor $448$
Sign $0.848 - 0.529i$
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  + (−0.608 − 0.793i)3-s + (−0.793 − 0.608i)5-s + (−0.258 + 0.965i)9-s + (−0.991 − 0.130i)11-s + (0.923 + 0.382i)13-s + i·15-s + (−0.866 − 0.5i)17-s + (−0.130 − 0.991i)19-s + (−0.258 + 0.965i)23-s + (0.258 + 0.965i)25-s + (0.923 − 0.382i)27-s + (−0.382 + 0.923i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.793 + 0.608i)37-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)3-s + (−0.793 − 0.608i)5-s + (−0.258 + 0.965i)9-s + (−0.991 − 0.130i)11-s + (0.923 + 0.382i)13-s + i·15-s + (−0.866 − 0.5i)17-s + (−0.130 − 0.991i)19-s + (−0.258 + 0.965i)23-s + (0.258 + 0.965i)25-s + (0.923 − 0.382i)27-s + (−0.382 + 0.923i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.793 + 0.608i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.848 - 0.529i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ 0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8108924625 - 0.2324727159i\)
\(L(\frac12)\) \(\approx\) \(0.8108924625 - 0.2324727159i\)
\(L(1)\) \(\approx\) \(0.6531261590 - 0.1933475929i\)
\(L(1)\) \(\approx\) \(0.6531261590 - 0.1933475929i\)

Euler product

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

−23.56763464852258809496875884796, −22.91900387131658110738533859377, −22.40412236648902851105890480028, −21.26724241857181562292025659127, −20.622773409256521871335476487426, −19.67816224615864485187695730208, −18.409195348631395647816028702391, −18.068369678696432250891660400768, −16.73304140434953189478837920217, −16.01031627153552984452380519076, −15.2636524125537110071219859748, −14.656303800090143766350273856015, −13.24461662465121636394851748391, −12.29566838536251912511906972646, −11.2072692119241006565651243728, −10.733145407655266375647592504834, −9.90063591484713586622685031202, −8.56106540198448553692648459894, −7.74149301691719577628982921675, −6.42782569272863542957928836599, −5.67984669975944607109705536496, −4.33612633570910928352953704350, −3.72327503358426037094644436393, −2.44985237270969219583861438452, −0.456090383165831825666249144796, 0.57524603389849208327274870190, 1.77445597839355360089503639784, 3.185873856254618456464610614333, 4.607803652274417876035995432293, 5.35550422506149971315569688708, 6.56696330197112405087988331027, 7.46061175140618142758118402415, 8.3077016639637478993438724193, 9.22049249723522403710432739425, 10.905525188790872811587906480487, 11.26672207931549460397397330307, 12.304806347259675178364107962674, 13.17640217989640987888748193653, 13.68784843335169083442857119695, 15.28563816089794929682264582818, 16.03832051042465830379975428445, 16.6911639072209400985656927979, 17.933465672022212503700505371328, 18.35989839464557528641312246880, 19.48999869298299198905913252089, 20.061072818438110145553377360586, 21.172439134391243625146339611649, 22.13277905389046166416220750635, 23.214690377407588393153951261184, 23.72377529339314307276528801572

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