Properties

Label 1-1400-1400.1227-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.729 - 0.684i$
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  + (−0.406 − 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

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.729 - 0.684i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.729 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099226093 - 0.4348689911i\)
\(L(\frac12)\) \(\approx\) \(1.099226093 - 0.4348689911i\)
\(L(1)\) \(\approx\) \(0.8900550211 - 0.2152737974i\)
\(L(1)\) \(\approx\) \(0.8900550211 - 0.2152737974i\)

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

−21.01036890142967983995321592940, −20.25717374808473360470642274944, −19.360955851208391699866697123502, −18.77224182709206907587551916715, −17.52617200737982082963504163723, −17.06112220755100054320643193689, −16.45475630901473582208839480046, −15.626134906478830401743598561901, −14.72011390915406205203538819684, −14.3925049821332138334380032639, −13.22624801118572391404508724202, −12.22629395380388293206662163655, −11.64249898827086851841327649800, −10.81150873914501382023929376293, −10.027563893455041311962691472760, −9.40689242191118242840718135304, −8.53096974745968837063294066198, −7.61991335606515069782199647745, −6.515485226713144391770918424683, −5.668894818691614516522871030999, −5.08669496408726487268810519483, −3.88970506887444026466005063409, −3.45739259706144666825127890587, −2.18589794818575136950141053716, −0.76558897015660209130610449032, 0.710515398325209150039820785300, 1.91249088352583765190718017071, 2.52776460934882746769729580634, 3.913183990749626150035690759927, 4.865286402361400400396332651, 5.69526418100494965693898115414, 6.72671691118981444068259253866, 7.14554583717390268561284918632, 8.06471854737217465320034169715, 8.957763146484038510116537664444, 9.90984020242488452776767653305, 10.74265355254721127045000092458, 11.69070153979718056431738791982, 12.38606359638994543962598000729, 12.73454096747412771035913550912, 13.951104306258086895862247027882, 14.44377749223312149341974386185, 15.27306231606569906159445948184, 16.48961239295058017092383541514, 17.044947953614186129425679777810, 17.59232504777757271263851178944, 18.5087550899208572655072086026, 19.17555908736200354942947600979, 19.75806038696758374840102784368, 20.58418070372214292056040053000

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