L-function 1-223-223.28-r0-0-0

• Label: 1-223-223.28-r0-0-0
• Degree: $1$
• Conductor: $223$
• Sign: $-0.305 - 0.952i$
• Analytic cond.: $1.03560$
• Root an. cond.: $1.03560$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.524 + 0.851i)2^-s + (−0.911 − 0.411i)3^-s + (−0.450 + 0.892i)4^-s + (−0.996 − 0.0848i)5^-s + (−0.127 − 0.991i)6^-s + (0.372 + 0.927i)7^-s + (−0.996 + 0.0848i)8^-s + (0.660 + 0.750i)9^-s + (−0.450 − 0.892i)10^-s + (−0.127 − 0.991i)11^-s + (0.778 − 0.628i)12^-s + (−0.911 + 0.411i)13^-s + (−0.594 + 0.803i)14^-s + (0.873 + 0.487i)15^-s + (−0.594 − 0.803i)16^-s + (−0.127 − 0.991i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$223$
Sign:	$-0.305 - 0.952i$
Analytic conductor:	$1.03560$
Root analytic conductor:	$1.03560$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{223} (28, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 223,\ (0:\ ),\ -0.305 - 0.952i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.03424742865 - 0.04692951972i$
$L(1)$	$\approx$	$0.5526853841 + 0.2244700788i$

Euler product

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

	$p$	$F_p(T)$
bad	223	$1$
good	2	$1 + (0.524 + 0.851i)T$
	3	$1 + (-0.911 - 0.411i)T$
	5	$1 + (-0.996 - 0.0848i)T$
	7	$1 + (0.372 + 0.927i)T$
	11	$1 + (-0.127 - 0.991i)T$
	13	$1 + (-0.911 + 0.411i)T$
	17	$1 + (-0.127 - 0.991i)T$
	19	$1 + (-0.721 - 0.691i)T$
	23	$1 + (-0.996 - 0.0848i)T$

Imaginary part of the first few zeros on the critical line

−27.22269522342802876955301888514, −26.21021279394698083128859420518, −24.27144339924900676196868398163, −23.61252606436184080141510281467, −22.98146572202880847599425335432, −22.22436167189524028813592326166, −21.19501257301375458526328715443, −20.18044727467260432053072734431, −19.61074682340562744963370339539, −18.26067804408694030060558010204, −17.41202296801820650992291003572, −16.29049607251395498935515588668, −15.0309606055242712250089755654, −14.55361581101866035887293330023, −12.69496384378583439625387159450, −12.38193998381389913972352717692, −11.117826389798647302846365381983, −10.566355345747575330716032522201, −9.65235475220955060806959957913, −7.891605310141999499005709494668, −6.708464540522863129271461072999, −5.26356002605501438970648509460, −4.306715885395680971634406275566, −3.69589442537302091257060784385, −1.72077963065749828087903754462, 0.03934758939018773510645279308, 2.57151604465234518862689194933, 4.24085540150002333687417666214, 5.138776429918033301935724335539, 6.102041966044927682896754627423, 7.23700811330821594394382912997, 8.05779426150454546267526218770, 9.170488420940560669453178079071, 11.153108034806590578982205849568, 11.84096957233558137020764235934, 12.55334943745162356753771317804, 13.72391141556272343161388355361, 14.91650876245115425202637185854, 15.84001491232406445591543374932, 16.48919140549705880072844381033, 17.51886587932505095099796449603, 18.53199415975255541366188073912, 19.24487827016131051726455938144, 20.925846540932275367319336643729, 22.09107117178282667772946933546, 22.39415393997094884308569729467, 23.770822512153994158763222639858, 24.11408355787692627341978736440, 24.790555902270009867627026265179, 26.14012045072168194302034715767

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