↑

Properties

Label 1-80e2-6400.6077-r1-0-0
Degree $1$
Conductor $6400$
Sign $0.998 - 0.0559i$
Analytic cond. $687.775$
Root an. cond. $687.775$
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  + (0.0588 + 0.998i)3-s + (0.195 + 0.980i)7-s + (−0.993 + 0.117i)9-s + (−0.976 − 0.214i)11-s + (−0.820 + 0.571i)13-s + (0.972 − 0.233i)17-s + (0.137 + 0.990i)19-s + (−0.967 + 0.252i)21-s + (−0.271 − 0.962i)23-s + (−0.175 − 0.984i)27-s + (0.747 + 0.664i)29-s + (0.156 − 0.987i)31-s + (0.156 − 0.987i)33-s + (−0.693 + 0.720i)37-s + (−0.619 − 0.785i)39-s + ⋯
L(s)  = 1  + (0.0588 + 0.998i)3-s + (0.195 + 0.980i)7-s + (−0.993 + 0.117i)9-s + (−0.976 − 0.214i)11-s + (−0.820 + 0.571i)13-s + (0.972 − 0.233i)17-s + (0.137 + 0.990i)19-s + (−0.967 + 0.252i)21-s + (−0.271 − 0.962i)23-s + (−0.175 − 0.984i)27-s + (0.747 + 0.664i)29-s + (0.156 − 0.987i)31-s + (0.156 − 0.987i)33-s + (−0.693 + 0.720i)37-s + (−0.619 − 0.785i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $0.998 - 0.0559i$
Analytic conductor: \(687.775\)
Root analytic conductor: \(687.775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6400} (6077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6400,\ (1:\ ),\ 0.998 - 0.0559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5384317285 + 0.01506914983i\)
\(L(\frac12)\) \(\approx\) \(0.5384317285 + 0.01506914983i\)
\(L(1)\) \(\approx\) \(0.7433419532 + 0.4153452380i\)
\(L(1)\) \(\approx\) \(0.7433419532 + 0.4153452380i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.0588 + 0.998i)T \)
7 \( 1 + (0.195 + 0.980i)T \)
11 \( 1 + (-0.976 - 0.214i)T \)
13 \( 1 + (-0.820 + 0.571i)T \)
17 \( 1 + (0.972 - 0.233i)T \)
19 \( 1 + (0.137 + 0.990i)T \)
23 \( 1 + (-0.271 - 0.962i)T \)
29 \( 1 + (0.747 + 0.664i)T \)
31 \( 1 + (0.156 - 0.987i)T \)
37 \( 1 + (-0.693 + 0.720i)T \)
41 \( 1 + (-0.785 - 0.619i)T \)
43 \( 1 + (-0.634 + 0.773i)T \)
47 \( 1 + (0.852 - 0.522i)T \)
53 \( 1 + (-0.505 + 0.862i)T \)
59 \( 1 + (0.0196 + 0.999i)T \)
61 \( 1 + (0.931 - 0.364i)T \)
67 \( 1 + (-0.252 + 0.967i)T \)
71 \( 1 + (-0.418 + 0.908i)T \)
73 \( 1 + (-0.872 - 0.488i)T \)
79 \( 1 + (-0.972 - 0.233i)T \)
83 \( 1 + (0.137 + 0.990i)T \)
89 \( 1 + (0.619 + 0.785i)T \)
97 \( 1 + (-0.156 - 0.987i)T \)
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

−17.64017404620757353011300661760, −17.07794557845300272344784064720, −16.16172787884336218278061449484, −15.48589612179000925158725818591, −14.68313155247601026839915271758, −14.06248224923973060359471045614, −13.47177387139472452759013470184, −12.97453695694170978604568480624, −12.2520312356372227794037947384, −11.69947146906635988681177508814, −10.83618932026072839191636934049, −10.236583375035751017964311712153, −9.64102034821886927607051723950, −8.5365528057088100232665132438, −7.97607840296286306550202884957, −7.35544529040883224443581652242, −7.027719005723070911096277284476, −6.05724214157043905803733490762, −5.21729880250822135916589026962, −4.801724186673802060777139600964, −3.54132095112986669410528001216, −2.99478876283137825115871441783, −2.148026943305298472397265618925, −1.3432762376105612077213957531, −0.52472510140864713371466722689, 0.11026020477920147899055957387, 1.46355392835789288759849889205, 2.568375654830630088022583855119, 2.82260671488158790719835663883, 3.8252873022639326948728765228, 4.62179792411254819800694461275, 5.29727835312146081691553805914, 5.67859802468799912130821587693, 6.57079216572949311864841741833, 7.62548366935025789400894064779, 8.28562341462395586950873338126, 8.771175151560941834970506533776, 9.626221509139013393877617826480, 10.173919719298980454541719539531, 10.615015443981559542932491497076, 11.76694701540904067583334496503, 11.92248816106006014185026500683, 12.72148043558429890022725496381, 13.7261678628307176472153995685, 14.40515619311332627107396834749, 14.811163955727106598229816178961, 15.51164734101244565505670482289, 16.14157262958629017232462308826, 16.60985813639170772039730059449, 17.271148197007122725057499954086

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