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Properties

Label 1-1400-1400.1363-r1-0-0
Degree $1$
Conductor $1400$
Sign $-0.986 - 0.165i$
Analytic cond. $150.450$
Root an. cond. $150.450$
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  + (−0.994 + 0.104i)3-s + (0.978 − 0.207i)9-s + (−0.978 − 0.207i)11-s + (0.951 + 0.309i)13-s + (0.406 + 0.913i)17-s + (−0.104 + 0.994i)19-s + (−0.743 − 0.669i)23-s + (−0.951 + 0.309i)27-s + (−0.809 + 0.587i)29-s + (0.913 − 0.406i)31-s + (0.994 + 0.104i)33-s + (−0.207 − 0.978i)37-s + (−0.978 − 0.207i)39-s + (−0.309 + 0.951i)41-s + i·43-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)3-s + (0.978 − 0.207i)9-s + (−0.978 − 0.207i)11-s + (0.951 + 0.309i)13-s + (0.406 + 0.913i)17-s + (−0.104 + 0.994i)19-s + (−0.743 − 0.669i)23-s + (−0.951 + 0.309i)27-s + (−0.809 + 0.587i)29-s + (0.913 − 0.406i)31-s + (0.994 + 0.104i)33-s + (−0.207 − 0.978i)37-s + (−0.978 − 0.207i)39-s + (−0.309 + 0.951i)41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.986 - 0.165i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01758674187 + 0.2108085281i\)
\(L(\frac12)\) \(\approx\) \(0.01758674187 + 0.2108085281i\)
\(L(1)\) \(\approx\) \(0.6939003715 + 0.08742072309i\)
\(L(1)\) \(\approx\) \(0.6939003715 + 0.08742072309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.994 + 0.104i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.743 - 0.669i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.207 - 0.978i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.406 - 0.913i)T \)
53 \( 1 + (-0.994 + 0.104i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.406 - 0.913i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.207 + 0.978i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−20.60583266831620761734320616189, −19.15067077354547982300785495319, −18.67374404422172310855418423866, −17.7514739460092869496865690810, −17.46529504051736611781609453421, −16.30774365377674613808664866682, −15.756070494438438199976709174679, −15.24980815782407546901655923746, −13.76180644025505957076293880104, −13.3780794936891516685781699443, −12.45588596925045456637175015723, −11.679327590442146387053552952631, −11.01334344032727016359983533017, −10.252644428594181253409698371738, −9.5320638122678006609567489181, −8.36141530607747529591386106269, −7.517044379252705954321575527464, −6.77343407804155169051965568099, −5.79106322558309964900344400419, −5.2138030147107169826435921712, −4.34198675626612893343356649630, −3.23049020116382196012381403312, −2.107116810164280098370936739644, −0.94671174876918636257455075888, −0.059001346806344997027381018101, 1.11708483408278774375294797658, 2.07032219510597451618431556898, 3.49828389270748560042109464346, 4.21301500944278894231831758411, 5.26403916113778324249431935452, 5.97280589369285917427947161856, 6.55889703839697995843712605580, 7.77327130875728820707662255384, 8.35337868834284841788937529110, 9.58663285024772901494041184851, 10.394921635103113302457399162579, 10.88509637577076105058401312222, 11.74429787096939392740286285672, 12.59980273333633011441695390638, 13.1263884971434644448551610753, 14.14608030401786267029333976925, 15.07680395140333947279290228606, 15.92272013981261650546643870301, 16.45829025285327284964377361291, 17.11229228112492215851438899572, 18.21448582767886904904131618387, 18.43603181774930014323291945811, 19.30342607092584753496693031792, 20.497962407753806103903491875332, 21.13623883697666044681164991582

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