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Properties

Label 1-448-448.213-r1-0-0
Degree $1$
Conductor $448$
Sign $-0.467 + 0.883i$
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  + (−0.793 − 0.608i)3-s + (−0.608 − 0.793i)5-s + (0.258 + 0.965i)9-s + (−0.130 − 0.991i)11-s + (−0.382 − 0.923i)13-s + i·15-s + (0.866 − 0.5i)17-s + (−0.991 − 0.130i)19-s + (−0.258 − 0.965i)23-s + (−0.258 + 0.965i)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 + (−0.608 − 0.793i)37-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)3-s + (−0.608 − 0.793i)5-s + (0.258 + 0.965i)9-s + (−0.130 − 0.991i)11-s + (−0.382 − 0.923i)13-s + i·15-s + (0.866 − 0.5i)17-s + (−0.991 − 0.130i)19-s + (−0.258 − 0.965i)23-s + (−0.258 + 0.965i)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 + (−0.608 − 0.793i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.467 + 0.883i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ -0.467 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2777785978 - 0.4612726762i\)
\(L(\frac12)\) \(\approx\) \(-0.2777785978 - 0.4612726762i\)
\(L(1)\) \(\approx\) \(0.5333158437 - 0.3732209753i\)
\(L(1)\) \(\approx\) \(0.5333158437 - 0.3732209753i\)

Euler product

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

−23.943609181614086880915533221796, −23.33680915439330629338134047467, −22.846889597692360015292579699421, −21.67532235698499383596847265772, −21.36136093830122098029992228087, −20.03101509176615221027203176168, −19.19997667103408445269087215441, −18.25251517604503163508655055295, −17.440635754633225605203164427208, −16.57506039359353348291947736314, −15.666043474132707402542378127958, −14.916096853638760812529145524835, −14.23025693853696877882659389302, −12.59525025527217384185653609214, −11.98059207988231466733244039224, −11.08682219147550157590907317790, −10.2435488504135429798363130442, −9.54077518527703614106744409701, −8.1354568364966308576430343264, −7.00332468345535596231340185453, −6.35155339727033297183506238891, −5.015984239121219960722208799678, −4.18217581464944356988837605920, −3.19981861751105446759538638330, −1.64236889729749909619634930400, 0.21786833327145720529176222598, 0.82833041721932895066505742474, 2.38912193800771209190590859282, 3.83862783433302689872807554461, 5.05570185056874665279866753057, 5.715902842209078848023099409489, 6.88516796811371454627012947379, 7.97884553334478213386611914463, 8.51554047306682409532736706335, 10.0272743313265918999248133233, 10.97134964496969136110787652388, 11.84537976967582633796110566170, 12.608344369324037559724022229539, 13.252025485132980973966877133773, 14.42075872630064660977366402976, 15.668015577313587900219668097845, 16.43955197134692493516601739140, 17.04212088264268645366112004878, 18.02890354264095567905627642928, 18.9950197840251300939514615077, 19.5689909032962232880970343575, 20.67003046571839606809789789364, 21.5460556453043340728878554766, 22.6320276281020057068013136246, 23.23693913982640766892933638836

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