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Properties

Label 1-1400-1400.1027-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.103 - 0.994i$
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  + (0.406 + 0.913i)3-s + (−0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (0.951 − 0.309i)13-s + (−0.994 + 0.104i)17-s + (0.913 + 0.406i)19-s + (−0.207 − 0.978i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (−0.104 − 0.994i)31-s + (−0.406 + 0.913i)33-s + (−0.743 − 0.669i)37-s + (0.669 + 0.743i)39-s + (−0.309 − 0.951i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)3-s + (−0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (0.951 − 0.309i)13-s + (−0.994 + 0.104i)17-s + (0.913 + 0.406i)19-s + (−0.207 − 0.978i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (−0.104 − 0.994i)31-s + (−0.406 + 0.913i)33-s + (−0.743 − 0.669i)37-s + (0.669 + 0.743i)39-s + (−0.309 − 0.951i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.103 - 0.994i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.103 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6327737452 - 0.5703749018i\)
\(L(\frac12)\) \(\approx\) \(0.6327737452 - 0.5703749018i\)
\(L(1)\) \(\approx\) \(1.042403580 + 0.2686267846i\)
\(L(1)\) \(\approx\) \(1.042403580 + 0.2686267846i\)

Euler product

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

−20.647514275467822067344004166962, −19.7477385919625970309048496714, −19.503848500933494447648648826852, −18.32150980632607037624822772325, −18.09507910257788227648816865671, −17.1136924711898529659203695173, −16.243324749038069928143990524624, −15.45881147112990451876346055748, −14.509517348247569986953988614068, −13.70254985842932112900559141470, −13.40142040194264902654589670736, −12.41538338079958852733301893368, −11.37653856874353994529371727974, −11.22287603195363381973399749570, −9.68918286509461177332277294021, −8.927753019401698683345806730808, −8.38426920656909554816427566554, −7.38351347480505177995288444660, −6.615832584019384151149717441617, −5.99696048475936868288820777074, −4.8970439442853487863204126472, −3.566987236849199474990248669588, −3.10877767500474831718369807033, −1.7244618871769440325535421286, −1.17532095533715454959774171287, 0.14728393306146539473534379871, 1.66215585681019855864787386622, 2.5714270579816763808608413960, 3.76928220058531692964517525007, 4.145413017570455325921682287799, 5.23203449788122533972246222699, 6.06098587123293487745426718588, 7.09883931517110029621862514235, 8.075804876894916479239902944417, 8.87369347941686371225614757051, 9.4936624928214444913204736254, 10.33752627402061015693324620957, 11.05093780297787728882277518351, 11.84229433864980357244114588132, 12.851652532939135946288634345366, 13.76223154454206992926131300524, 14.34349960140021544881616372333, 15.348574861212354002700137196720, 15.621331877406247155327004092622, 16.645983404616143505479426516887, 17.24235957675596004375090834401, 18.19898662689762967365661546957, 18.95843778181602308653524587672, 20.09951474920999871367183597219, 20.30382626009609167473507264689

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