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Properties

Label 1-4400-4400.131-r0-0-0
Degree $1$
Conductor $4400$
Sign $0.990 + 0.140i$
Analytic cond. $20.4335$
Root an. cond. $20.4335$
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  + (−0.951 − 0.309i)3-s − 7-s + (0.809 + 0.587i)9-s + (−0.587 + 0.809i)13-s + (−0.309 − 0.951i)17-s + (−0.951 + 0.309i)19-s + (0.951 + 0.309i)21-s + (−0.809 + 0.587i)23-s + (−0.587 − 0.809i)27-s + (−0.951 − 0.309i)29-s + (−0.309 − 0.951i)31-s + (−0.587 + 0.809i)37-s + (0.809 − 0.587i)39-s + (−0.809 − 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)3-s − 7-s + (0.809 + 0.587i)9-s + (−0.587 + 0.809i)13-s + (−0.309 − 0.951i)17-s + (−0.951 + 0.309i)19-s + (0.951 + 0.309i)21-s + (−0.809 + 0.587i)23-s + (−0.587 − 0.809i)27-s + (−0.951 − 0.309i)29-s + (−0.309 − 0.951i)31-s + (−0.587 + 0.809i)37-s + (0.809 − 0.587i)39-s + (−0.809 − 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $0.990 + 0.140i$
Analytic conductor: \(20.4335\)
Root analytic conductor: \(20.4335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4400} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4400,\ (0:\ ),\ 0.990 + 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3920915950 + 0.02776157387i\)
\(L(\frac12)\) \(\approx\) \(0.3920915950 + 0.02776157387i\)
\(L(1)\) \(\approx\) \(0.5377296194 - 0.03917557612i\)
\(L(1)\) \(\approx\) \(0.5377296194 - 0.03917557612i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 - T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.951 - 0.309i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (0.587 - 0.809i)T \)
61 \( 1 + (0.587 + 0.809i)T \)
67 \( 1 + (0.951 - 0.309i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−18.20012871251016780207506868974, −17.44433041660767593708795773307, −16.996209449865126201004239141560, −16.26732802959076770590628740949, −15.73834439601073372515820088712, −15.02644259405022580941917426914, −14.4428212146136480348924665270, −13.15402746439517321549819014114, −12.81055328308936898087140478890, −12.312009642701794932877201663463, −11.433058170274662396040498029487, −10.566702819539977475974443267508, −10.29690269334647147665407837161, −9.52419778305527882322420085799, −8.73969179725757250233415911906, −7.870996880262008958433249711048, −6.835002458707671021972504234995, −6.49477039327569362323569288747, −5.66800116946514276425609862467, −5.06352045348682826228194963445, −4.093795988423035501938203763, −3.5599013034950681146381107340, −2.53545788719077861008714280553, −1.55335392740191637995372529392, −0.29688652814926654129583513775, 0.3834364995096303362031753394, 1.76478419106179032921927469163, 2.31114154118858867450503631525, 3.52128285214082277138874656898, 4.23888387904412816459517291867, 5.04638160090665722485359287588, 5.83246036159451516909782461303, 6.47179975041017378694540778702, 7.0608376275266414071280063760, 7.68508740819888304442195272384, 8.75306060697034200849272835695, 9.66313622206235889217649970005, 9.96590306534066003599795252723, 10.92156140755795626716985229071, 11.61802186591049585888932800745, 12.12187774956396721286181648827, 12.85628018422539198961256462409, 13.429023180549236368121710015214, 14.10298495331486788982112534796, 15.107188192554149485593132642056, 15.83992587269591521738466035522, 16.32275533466551024417009049512, 17.09066661341142222704726797786, 17.37878849127784845768930471341, 18.498436146798507868473513456945

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