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Properties

Label 1-224-224.125-r1-0-0
Degree $1$
Conductor $224$
Sign $0.555 - 0.831i$
Analytic cond. $24.0721$
Root an. cond. $24.0721$
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  + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + i·9-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)13-s − 15-s + 17-s + (0.707 + 0.707i)19-s + i·23-s − i·25-s + (0.707 − 0.707i)27-s + (0.707 + 0.707i)29-s − 31-s − 33-s + (−0.707 + 0.707i)37-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + i·9-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)13-s − 15-s + 17-s + (0.707 + 0.707i)19-s + i·23-s − i·25-s + (0.707 − 0.707i)27-s + (0.707 + 0.707i)29-s − 31-s − 33-s + (−0.707 + 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(24.0721\)
Root analytic conductor: \(24.0721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 224,\ (1:\ ),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.687922890 - 0.9022135813i\)
\(L(\frac12)\) \(\approx\) \(1.687922890 - 0.9022135813i\)
\(L(1)\) \(\approx\) \(1.090653061 - 0.3423136927i\)
\(L(1)\) \(\approx\) \(1.090653061 - 0.3423136927i\)

Euler product

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

−26.36183903316331298877948284254, −25.6005508799565157627557449415, −24.61324622149710807793476073165, −23.11961477822167127214061709131, −22.707191619157735283179954850349, −21.791918527475954285533867737906, −20.95677732953237638368512096881, −20.00742357834492830816332466257, −18.509779870710357730713370725276, −17.79816896181893701862380561872, −17.007888430545665019520233811, −15.925110603777765645186614585885, −14.932339345573583710069607292251, −14.13511569675605439343991055785, −12.77599374436755463275234724791, −11.6930621055830241596114974573, −10.66657247304937255467507501974, −9.94864375145313827549521650171, −9.001057924516890082460591485072, −7.28821206932803402186483241693, −6.22455425236326142676177184142, −5.38538541223822912809115688426, −4.05519217915864634290231738239, −2.837847846619779736004870369970, −1.05288190967685616543022269819, 0.95299030507813112415699471745, 1.75139763427535185238706849017, 3.65506076335518427848451754793, 5.277755541488717307565487393408, 5.91161495991977525985819205879, 7.030469073080084462774895536713, 8.32685185947896981852459829304, 9.35144838589790291580498874134, 10.597042158464399075601508853268, 11.76165226113779201127198273834, 12.43498661026730675003155714836, 13.683011735915425334182390621013, 14.10726863634340645879057237276, 16.0658624618889852045331733589, 16.66521068275058744172135320407, 17.4967050010898594330120088529, 18.50254690770067047044976065318, 19.29116898734253543975824024162, 20.53926817429235575475356655144, 21.52503582733543381799015383043, 22.317757982817606376366154149730, 23.54430433372540526104298840998, 24.10896060964159892305776667398, 25.11587078019996653718728643630, 25.7086997470811618188178244793

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