L-function 1-405-405.88-r1-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
good	2	$1 + (0.957 + 0.286i)T$
	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	$1 + (0.893 - 0.448i)T$

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