L-function 1-1125-1125.391-r0-0-0

• Label: 1-1125-1125.391-r0-0-0
• Degree: $1$
• Conductor: $1125$
• Sign: $0.896 - 0.442i$
• Analytic cond.: $5.22447$
• Root an. cond.: $5.22447$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.699 + 0.714i)2^-s + (−0.0209 − 0.999i)4^-s + (−0.104 − 0.994i)7^-s + (0.728 + 0.684i)8^-s + (−0.699 + 0.714i)11^-s + (0.146 + 0.989i)13^-s + (0.783 + 0.621i)14^-s + (−0.999 + 0.0418i)16^-s + (0.876 − 0.481i)17^-s + (−0.187 − 0.982i)19^-s + (−0.0209 − 0.999i)22^-s + (0.832 − 0.553i)23^-s + (−0.809 − 0.587i)26^-s + (−0.992 + 0.125i)28^-s + (0.387 − 0.921i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1125$    =    $3^{2} \cdot 5^{3}$
Sign:	$0.896 - 0.442i$
Analytic conductor:	$5.22447$
Root analytic conductor:	$5.22447$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1125} (391, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1125,\ (0:\ ),\ 0.896 - 0.442i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8537486905 - 0.1989877220i$
$L(1)$	$\approx$	$0.7342182759 + 0.07436976039i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
good	2	$1 + (-0.699 + 0.714i)T$
	7	$1 + (-0.104 - 0.994i)T$
	11	$1 + (-0.699 + 0.714i)T$
	13	$1 + (0.146 + 0.989i)T$
	17	$1 + (0.876 - 0.481i)T$
	19	$1 + (-0.187 - 0.982i)T$
	23	$1 + (0.832 - 0.553i)T$
	29	$1 + (0.387 - 0.921i)T$
	31	$1 + (-0.855 - 0.518i)T$

Imaginary part of the first few zeros on the critical line

−21.44674160988394509430345214640, −20.61805136095396072505801275328, −19.81368717536164312129871212669, −18.85706365794764122898653270313, −18.57737360988408179833524006779, −17.76128200827328632361720149934, −16.85215752417966595076176691999, −16.10021615368093656296102100470, −15.3757018404283817825041530364, −14.37647439799588358949633964001, −13.23527495898863114370846745652, −12.5676132233221067543785025131, −12.02939643813723150404101522784, −10.87746361849005757635680761673, −10.46921282235614210157919784546, −9.46670273269116663596146809552, −8.6038928100414754103009341357, −8.086541061362092856621129403301, −7.15669606894015829784706720125, −5.79737432354530203217258491389, −5.24629184147766848740864095525, −3.63712821559954364321527751873, −3.10897875765849272325257842689, −2.11757065125457149544076786931, −1.00406469485695521152757939954, 0.56056841120251279352720423334, 1.69277601173625708728649855960, 2.87567176612297604374927867629, 4.39596455218428632476554941976, 4.88212942441986386858394264833, 6.12687128157302149606691922777, 6.971518452076323182507911874072, 7.4957195477144183356180304542, 8.3663901854344146853219050288, 9.45740873369882335338683855719, 9.91941917517450112024221622365, 10.86435817617081444380846505626, 11.532419080526455650052909507199, 12.896094867816705410453357680370, 13.6120106728576561276490611626, 14.42071608208784324837328609867, 15.14229606741243741913256475778, 16.05379599961238947217304098592, 16.68466574219661844409750706148, 17.31552699309036237715399245603, 18.129069845214816429280248567947, 18.90591107459016628067997810860, 19.548944646474774461642809186406, 20.50677405214995765021260688621, 20.96323296948546459182261294378

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