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Properties

Label 1-448-448.403-r1-0-0
Degree $1$
Conductor $448$
Sign $-0.935 - 0.354i$
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  + (−0.991 + 0.130i)3-s + (0.130 − 0.991i)5-s + (0.965 − 0.258i)9-s + (−0.608 + 0.793i)11-s + (−0.923 − 0.382i)13-s + i·15-s + (0.866 − 0.5i)17-s + (0.793 − 0.608i)19-s + (0.965 − 0.258i)23-s + (−0.965 − 0.258i)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 + (−0.130 + 0.991i)37-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)3-s + (0.130 − 0.991i)5-s + (0.965 − 0.258i)9-s + (−0.608 + 0.793i)11-s + (−0.923 − 0.382i)13-s + i·15-s + (0.866 − 0.5i)17-s + (0.793 − 0.608i)19-s + (0.965 − 0.258i)23-s + (−0.965 − 0.258i)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 + (−0.130 + 0.991i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.935 - 0.354i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ -0.935 - 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1171029109 - 0.6398085494i\)
\(L(\frac12)\) \(\approx\) \(0.1171029109 - 0.6398085494i\)
\(L(1)\) \(\approx\) \(0.6999138622 - 0.1861371063i\)
\(L(1)\) \(\approx\) \(0.6999138622 - 0.1861371063i\)

Euler product

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

−23.97981962626063278550694315339, −23.256830790761262987949003143712, −22.52368816348533717600670070065, −21.625096920523941054944251497683, −21.22445846712724405185483244027, −19.60384081898162958963797252164, −18.79315276557774390205314643427, −18.24591962246112707265964385794, −17.27517497318683401439860942007, −16.49181460892252026625909188066, −15.61485729928055076885315989965, −14.53081443903647659340154400852, −13.756188652378215691391988643, −12.536735402947859921629103206, −11.8426620138083805953774130383, −10.68457264450463649073895513192, −10.417606545295293685115750336654, −9.14372752442480358343938036731, −7.55230696696403068509570595967, −7.08749037973749913771955703352, −5.826527336267760472235716702025, −5.29373589758956026560492602644, −3.800840467167068905427317377784, −2.68886008129436713036116971612, −1.2691729100283196372899017940, 0.225887901794479214746680393892, 1.23117079106708396034388330525, 2.735244458622054860429323215380, 4.46077935472787138030336247685, 5.01451993469330438593186028899, 5.81759421563674086239605580673, 7.16951054650963788897436165146, 7.92006777000619193778304075508, 9.52199252752413743452069132595, 9.84211269659773778517244831900, 11.115593405170921510280460901669, 12.079750779152970329635716714184, 12.6645629482263887682586009376, 13.49939764739484841806249362834, 14.94666492205123605392019473239, 15.75157556897901173785195642831, 16.614839862196812756843687652675, 17.30164060712890394277888062075, 17.993124828538061758219468862631, 19.02998335353621133776281772634, 20.22292075118673451932826569240, 20.870344582194062860819056364119, 21.71543220365045980183416272943, 22.71921303981070140997477494110, 23.29897670239885860307007130928

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