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Properties

Label 1-1400-1400.291-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.729 + 0.684i$
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  + (0.913 + 0.406i)3-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.309 + 0.951i)13-s + (−0.104 + 0.994i)17-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.104 − 0.994i)31-s + (0.913 − 0.406i)33-s + (−0.669 − 0.743i)37-s + (−0.669 + 0.743i)39-s + (0.309 − 0.951i)41-s + 43-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)3-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.309 + 0.951i)13-s + (−0.104 + 0.994i)17-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.104 − 0.994i)31-s + (0.913 − 0.406i)33-s + (−0.669 − 0.743i)37-s + (−0.669 + 0.743i)39-s + (0.309 − 0.951i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.729 + 0.684i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.729 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.532041898 + 1.397324450i\)
\(L(\frac12)\) \(\approx\) \(3.532041898 + 1.397324450i\)
\(L(1)\) \(\approx\) \(1.643462529 + 0.3156738576i\)
\(L(1)\) \(\approx\) \(1.643462529 + 0.3156738576i\)

Euler product

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

−20.32197821903712483817522102329, −19.917119249048841832385951153, −19.117092326016535206180887800240, −18.19755652152121464654577297484, −17.765335504746797709456851450955, −16.783884891644966975548686855009, −15.7058385505522654579927754539, −15.20802336629205432351914865009, −14.2235551465623557840341939693, −13.90952760852390457427411346289, −12.708894010090741273814633401103, −12.37780485624435183385878216191, −11.385907336251730856844450185948, −10.209973486641929373744901393549, −9.5796936710994151637217501561, −8.82532132624262999572731245223, −7.9638045968479930142640755968, −7.16464394219557281652125176072, −6.629853822437512882897687645394, −5.283279360944995141919660785729, −4.49984144346987509722104779937, −3.27489884404870946825069476432, −2.80800084402993060947011389897, −1.58605160102073505454678984721, −0.780363403909400511683930641068, 0.93001846355281290064204579196, 1.9875320772781417914346675132, 2.92407332468863133321444213437, 3.8290119159911837464999645100, 4.467228726949815582890641807870, 5.56851958107156174079735402554, 6.6046273460945411338372650669, 7.45043706254637900804504589812, 8.319500487292157203818515505612, 9.14794024317132000555866992907, 9.53599034631057647711894129061, 10.66922211619012413582540761886, 11.32975886615542398961864528458, 12.314355901844100742978085918332, 13.24913316131289075162211786108, 14.03317030783273711766595217048, 14.43586516542696575246967658332, 15.42163443782718026113104317644, 16.00051229332336639419132022684, 16.892819968703767727267781336, 17.53714478808833650589725415086, 18.844341345493948729534611230312, 19.2463667718874101627024132745, 19.78976136075475563726271350877, 20.86231073565353973802719394143

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