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Properties

Label 1-1400-1400.717-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.0847 - 0.996i$
Analytic cond. $6.50157$
Root an. cond. $6.50157$
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  + (0.207 + 0.978i)3-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (−0.587 + 0.809i)13-s + (−0.743 − 0.669i)17-s + (0.978 + 0.207i)19-s + (−0.994 + 0.104i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.669 + 0.743i)31-s + (0.207 − 0.978i)33-s + (−0.406 − 0.913i)37-s + (−0.913 − 0.406i)39-s + (0.809 + 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)3-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (−0.587 + 0.809i)13-s + (−0.743 − 0.669i)17-s + (0.978 + 0.207i)19-s + (−0.994 + 0.104i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.669 + 0.743i)31-s + (0.207 − 0.978i)33-s + (−0.406 − 0.913i)37-s + (−0.913 − 0.406i)39-s + (0.809 + 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0847 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0847 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.0847 - 0.996i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.0847 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3139606663 - 0.2883869542i\)
\(L(\frac12)\) \(\approx\) \(0.3139606663 - 0.2883869542i\)
\(L(1)\) \(\approx\) \(0.7958172299 + 0.1935744690i\)
\(L(1)\) \(\approx\) \(0.7958172299 + 0.1935744690i\)

Euler product

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

−20.54722690769587710951129774885, −20.28994530620584574080367135200, −19.477767205703863765382815028606, −18.66300015112293384109593947991, −17.85486715217725476625787245308, −17.60349321707650284229351715966, −16.49859053146875123780076518828, −15.51118003740642131443752850850, −14.91995623952562699040377411053, −13.93609014420712232229831372577, −13.328268936523939715919153683437, −12.52397565336564516166646269347, −12.06193218887765347162488273255, −10.93812339687529276928768650395, −10.22794549355029180597549737488, −9.19349123278580334520073445015, −8.327636116383302631376303834571, −7.57130424708210190714218918495, −7.054626442202649485761127724809, −5.898637707997799974909225310688, −5.30986483778109219953133920582, −4.10271655477895153559711607402, −2.89952038773740194552830944239, −2.30875599948422495285916848829, −1.20585784503237603994754654812, 0.155717727787021302146332618877, 2.01309974364065780944107462573, 2.82862015489330432066756907021, 3.75534190773762681118907795774, 4.67349831398054553944658773953, 5.31420865270833808116250977319, 6.25483520526750115008522904515, 7.434288269244948377707690779218, 8.15048629310567521326879042472, 9.15101035825725822329810893086, 9.68187896515466855784873526243, 10.50834851492905836875175452668, 11.31580774434911232706325532514, 11.95352987434309028308971433552, 13.11001748863424200083721175553, 14.01670348224759414565956721926, 14.38734727461262891087692901887, 15.55625935804243647390778696893, 15.9998917016732083194606921263, 16.56701780753248839433433189839, 17.62719230946285257761550219971, 18.26502007144885633876316651226, 19.30143101842979191239998410397, 19.95417032867363773732273413745, 20.685773251459551712323966263855

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