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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.8537486905 - 0.1989877220i\)
\(L(\frac12)\) \(\approx\) \(0.8537486905 - 0.1989877220i\)
\(L(1)\) \(\approx\) \(0.7342182759 + 0.07436976039i\)
\(L(1)\) \(\approx\) \(0.7342182759 + 0.07436976039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.699 + 0.714i)T \)
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 \( 1 + (-0.855 - 0.518i)T \)
37 \( 1 + (0.535 + 0.844i)T \)
41 \( 1 + (-0.895 + 0.444i)T \)
43 \( 1 + (0.669 + 0.743i)T \)
47 \( 1 + (0.228 + 0.973i)T \)
53 \( 1 + (-0.425 - 0.904i)T \)
59 \( 1 + (0.985 + 0.166i)T \)
61 \( 1 + (-0.895 - 0.444i)T \)
67 \( 1 + (0.387 + 0.921i)T \)
71 \( 1 + (0.728 - 0.684i)T \)
73 \( 1 + (-0.637 - 0.770i)T \)
79 \( 1 + (-0.756 - 0.653i)T \)
83 \( 1 + (0.944 + 0.328i)T \)
89 \( 1 + (-0.637 - 0.770i)T \)
97 \( 1 + (0.387 - 0.921i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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