↑

Properties

Label 1-1400-1400.1053-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.937 - 0.347i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.743 + 0.669i)3-s + (0.104 + 0.994i)9-s + (0.104 − 0.994i)11-s + (−0.587 − 0.809i)13-s + (−0.207 − 0.978i)17-s + (−0.669 − 0.743i)19-s + (0.406 + 0.913i)23-s + (−0.587 + 0.809i)27-s + (0.309 + 0.951i)29-s + (0.978 − 0.207i)31-s + (0.743 − 0.669i)33-s + (0.994 − 0.104i)37-s + (0.104 − 0.994i)39-s + (0.809 − 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)3-s + (0.104 + 0.994i)9-s + (0.104 − 0.994i)11-s + (−0.587 − 0.809i)13-s + (−0.207 − 0.978i)17-s + (−0.669 − 0.743i)19-s + (0.406 + 0.913i)23-s + (−0.587 + 0.809i)27-s + (0.309 + 0.951i)29-s + (0.978 − 0.207i)31-s + (0.743 − 0.669i)33-s + (0.994 − 0.104i)37-s + (0.104 − 0.994i)39-s + (0.809 − 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.937 - 0.347i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.937 - 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.845980358 - 0.3311089107i\)
\(L(\frac12)\) \(\approx\) \(1.845980358 - 0.3311089107i\)
\(L(1)\) \(\approx\) \(1.318373241 + 0.05724957234i\)
\(L(1)\) \(\approx\) \(1.318373241 + 0.05724957234i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−20.86244091728397295073727354947, −19.93629468154976395222465895141, −19.30786345404308819170500535731, −18.80329910085910413156396479684, −17.80450937822709578956458660561, −17.24976434063087506065211106115, −16.3632104880010978758110389113, −15.16957067516543553026446444343, −14.7657471650969654798916015629, −14.07899140973882689759175074159, −13.104075793564917804886993264846, −12.4840025669903470978899949868, −11.91784267939161452119589694486, −10.75549957945445422100946938059, −9.78759444772621291154114482533, −9.1997687872992365122672379740, −8.16681471737729145224797297418, −7.68217348828951558547632189470, −6.55830898896739636815357150769, −6.22567919360754815001152259824, −4.55763058789343652959044059295, −4.127181758932884904472804318564, −2.76677338415168409677640342723, −2.12126459995500655009702036039, −1.18695454075976092222105127175, 0.69738237939354831953127794402, 2.25781436207479255660668497952, 2.97412307577536003105380405094, 3.741031623623208166896200024, 4.84456779881571510358441275784, 5.404219660389931095105695864739, 6.6426313771447941355589376261, 7.59187445570023470657908512341, 8.350782982006693028333226079482, 9.12558164659030714231384537479, 9.76059179771478481120583253275, 10.7522022752302828334435201129, 11.252876837475377290882370546877, 12.39647036736433044340166440224, 13.381392833253021921375237651807, 13.85701343856617492990414926047, 14.74255855040357117924599712821, 15.45626088382600901341787899966, 16.03572310301113299732935983918, 16.90150923644392764583239910480, 17.65298877893533010529602393304, 18.6895640820909824318842972737, 19.41937616125896129552275950234, 19.980700037956990354862003134070, 20.72850942643624867375002073731

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