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Properties

Label 1-224-224.3-r0-0-0
Degree $1$
Conductor $224$
Sign $0.945 - 0.325i$
Analytic cond. $1.04025$
Root an. cond. $1.04025$
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  + (0.258 − 0.965i)3-s + (0.258 + 0.965i)5-s + (−0.866 − 0.5i)9-s + (0.965 + 0.258i)11-s + (0.707 + 0.707i)13-s + 15-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.258 − 0.965i)37-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (0.258 + 0.965i)5-s + (−0.866 − 0.5i)9-s + (0.965 + 0.258i)11-s + (0.707 + 0.707i)13-s + 15-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.258 − 0.965i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.945 - 0.325i$
Analytic conductor: \(1.04025\)
Root analytic conductor: \(1.04025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 224,\ (0:\ ),\ 0.945 - 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.386753341 - 0.2319103647i\)
\(L(\frac12)\) \(\approx\) \(1.386753341 - 0.2319103647i\)
\(L(1)\) \(\approx\) \(1.224989134 - 0.1644639283i\)
\(L(1)\) \(\approx\) \(1.224989134 - 0.1644639283i\)

Euler product

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

−26.7270505775167105825096394191, −25.36970063594098747531010778589, −24.96490950222856539975019320652, −23.739931798750841950172440448884, −22.56333130723738843644858862369, −21.78258220720471410489443304728, −20.80437646393530538957769155556, −20.18464834873329069724367377022, −19.29337536619488846499626032441, −17.7198710607508269282969169038, −16.89562841789038744770041964968, −16.086289270482695894229624272200, −15.19376056276363744627125210429, −14.08547820967951787078758736116, −13.16627358990592458429752519701, −11.94796492564883897559050014007, −10.8400471827579640922751403842, −9.786309896783534173872285129694, −8.83727467540261594213847708208, −8.19184778076462030654454067786, −6.30843142897406884761813581105, −5.24671316212931778548721898304, −4.22394957244049640375452366913, −3.14620332113865483592078744829, −1.33775026221339903875275151509, 1.402606467298128643609094942819, 2.62110428928483527268969825834, 3.75663631933978617675876219881, 5.59329164806854491966540484002, 6.85632798352005351661341456910, 7.14891569346581617461289282795, 8.741132959176112512889756478319, 9.59983317802879496827015407877, 11.2124124531795996962657388251, 11.72338561486231768017557127376, 13.131066391586331163475749255399, 13.995632040803056161293160470483, 14.6003866569338420347537586673, 15.884702162410360823911250431731, 17.26898292231761294822383051606, 18.05522809716177870199541270459, 18.80496547137179902227547052700, 19.665580200834325324068305906691, 20.64303287169052027213243965432, 21.939729098528349723998651532720, 22.762221375198354870181960437097, 23.58883475037916192164817117046, 24.72524940957652218013453985682, 25.38430078547244037173049785088, 26.25873918751142016915819602307

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