↑

Properties

Label 1-405-405.88-r1-0-0
Degree $1$
Conductor $405$
Sign $-0.222 - 0.975i$
Analytic cond. $43.5232$
Root an. cond. $43.5232$
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  + (0.957 + 0.286i)2-s + (0.835 + 0.549i)4-s + (−0.998 + 0.0581i)7-s + (0.642 + 0.766i)8-s + (0.597 − 0.802i)11-s + (−0.727 − 0.686i)13-s + (−0.973 − 0.230i)14-s + (0.396 + 0.918i)16-s + (−0.984 − 0.173i)17-s + (−0.173 − 0.984i)19-s + (0.802 − 0.597i)22-s + (−0.998 − 0.0581i)23-s + (−0.5 − 0.866i)26-s + (−0.866 − 0.5i)28-s + (−0.973 + 0.230i)29-s + ⋯
L(s)  = 1  + (0.957 + 0.286i)2-s + (0.835 + 0.549i)4-s + (−0.998 + 0.0581i)7-s + (0.642 + 0.766i)8-s + (0.597 − 0.802i)11-s + (−0.727 − 0.686i)13-s + (−0.973 − 0.230i)14-s + (0.396 + 0.918i)16-s + (−0.984 − 0.173i)17-s + (−0.173 − 0.984i)19-s + (0.802 − 0.597i)22-s + (−0.998 − 0.0581i)23-s + (−0.5 − 0.866i)26-s + (−0.866 − 0.5i)28-s + (−0.973 + 0.230i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.222 - 0.975i$
Analytic conductor: \(43.5232\)
Root analytic conductor: \(43.5232\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (1:\ ),\ -0.222 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9656821711 - 1.210308263i\)
\(L(\frac12)\) \(\approx\) \(0.9656821711 - 1.210308263i\)
\(L(1)\) \(\approx\) \(1.418184708 + 0.007523746248i\)
\(L(1)\) \(\approx\) \(1.418184708 + 0.007523746248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.957 + 0.286i)T \)
7 \( 1 + (-0.998 + 0.0581i)T \)
11 \( 1 + (0.597 - 0.802i)T \)
13 \( 1 + (-0.727 - 0.686i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.998 - 0.0581i)T \)
29 \( 1 + (-0.973 + 0.230i)T \)
31 \( 1 + (0.893 - 0.448i)T \)
37 \( 1 + (0.342 + 0.939i)T \)
41 \( 1 + (-0.286 - 0.957i)T \)
43 \( 1 + (0.116 - 0.993i)T \)
47 \( 1 + (0.448 - 0.893i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.597 - 0.802i)T \)
61 \( 1 + (-0.835 + 0.549i)T \)
67 \( 1 + (0.230 - 0.973i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.642 + 0.766i)T \)
79 \( 1 + (0.286 - 0.957i)T \)
83 \( 1 + (-0.957 - 0.286i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.918 - 0.396i)T \)
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

−24.32110174907477252186368218964, −23.27890113748222628864982367309, −22.567595394888370130525605936159, −21.98739626971802519756602870179, −21.03273963483410631590347912373, −19.89554817021561570021940406530, −19.60701439985498717179295887709, −18.50560993692448719543794355656, −17.14649595945506848535544495345, −16.30964913784023454208180958652, −15.42603055220749291930757023417, −14.53665047186236701061373151077, −13.7388502981584190755203649861, −12.65904458416650171115749637352, −12.18332361921971447000099124388, −11.10808334208228562506081712708, −9.956311352843453655375965061811, −9.37174670914552085492935743209, −7.64366463382911942462294320770, −6.63028281148976580220256127885, −6.00734827830228014326217694557, −4.54834789903756526218785685672, −3.91160583170365605871425175699, −2.62773846313718282800081919238, −1.61972893567548311181315410041, 0.26803134965195317735241378667, 2.25724986951486034495013151419, 3.21140615691019477511877112872, 4.18304520575271209980967186548, 5.36406121493060922180004629337, 6.32942767291516554393919933140, 7.026909602322748695828672133480, 8.24893852058615705051581812323, 9.36168882025363766572605983787, 10.56625927410678926566983117963, 11.567482413212207684105528257429, 12.437695616748073990935418443365, 13.337558632760974708881411395076, 13.940383732003143810966888917160, 15.206117760673852883894845956324, 15.69612780714883674883384106515, 16.74687040601027374339011852347, 17.40546667124355320975916093291, 18.83243300277132471257771963935, 19.84145984118266695784723045048, 20.30020969836564671616221117, 21.77148455049540030685159326972, 22.14665773529545907154933594105, 22.79532870520046360526632169964, 24.06048267001176077941334795427

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