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Properties

Label 1-2205-2205.157-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.624 - 0.780i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (−0.680 − 0.733i)2-s + (−0.0747 + 0.997i)4-s + (0.781 − 0.623i)8-s + (−0.222 − 0.974i)11-s + (−0.680 − 0.733i)13-s + (−0.988 − 0.149i)16-s + (0.997 − 0.0747i)17-s + (−0.5 + 0.866i)19-s + (−0.563 + 0.826i)22-s + (0.433 + 0.900i)23-s + (−0.0747 + 0.997i)26-s + (−0.826 + 0.563i)29-s + (0.5 − 0.866i)31-s + (0.563 + 0.826i)32-s + (−0.733 − 0.680i)34-s + ⋯
L(s)  = 1  + (−0.680 − 0.733i)2-s + (−0.0747 + 0.997i)4-s + (0.781 − 0.623i)8-s + (−0.222 − 0.974i)11-s + (−0.680 − 0.733i)13-s + (−0.988 − 0.149i)16-s + (0.997 − 0.0747i)17-s + (−0.5 + 0.866i)19-s + (−0.563 + 0.826i)22-s + (0.433 + 0.900i)23-s + (−0.0747 + 0.997i)26-s + (−0.826 + 0.563i)29-s + (0.5 − 0.866i)31-s + (0.563 + 0.826i)32-s + (−0.733 − 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.624 - 0.780i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9070257637 - 0.4359737913i\)
\(L(\frac12)\) \(\approx\) \(0.9070257637 - 0.4359737913i\)
\(L(1)\) \(\approx\) \(0.7196083676 - 0.2380167553i\)
\(L(1)\) \(\approx\) \(0.7196083676 - 0.2380167553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.680 - 0.733i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.680 - 0.733i)T \)
17 \( 1 + (0.997 - 0.0747i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.433 + 0.900i)T \)
29 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.563 + 0.826i)T \)
41 \( 1 + (-0.365 + 0.930i)T \)
43 \( 1 + (0.930 - 0.365i)T \)
47 \( 1 + (0.680 + 0.733i)T \)
53 \( 1 + (0.563 - 0.826i)T \)
59 \( 1 + (0.365 + 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.294 + 0.955i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.680 - 0.733i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−19.4810522223573138394774833849, −19.15474603047474676787773443829, −18.277456158571386899368975913360, −17.60850672380534797765133625384, −16.94120123641851711241744009187, −16.406721612124799951487654305273, −15.48274005848720940195584511650, −14.875318591685757773214114395324, −14.338646215232653896523848656900, −13.453555101907885220457772739744, −12.51305715760773899337096687201, −11.78979387184716237870177456498, −10.709214587291373077351619499894, −10.21121530580572051827412205418, −9.30661018504621682541079204748, −8.86189840817693465110238402098, −7.74954521690584466953083875080, −7.243203366065168350546290009582, −6.55705695617335031132530639288, −5.600701598157026869100722720696, −4.80544742491127573612478729820, −4.13861847946292058894255313727, −2.59658562592600116309047978790, −1.90857148485698607985961542108, −0.696990121917753286999094612223, 0.67614841381307600806862069550, 1.583725862770626585136786590067, 2.7257374564061136337928669145, 3.28816230288666981140742934183, 4.16006727016390911537890361728, 5.29773723385876536068117516162, 6.03892327874173852511146551922, 7.27998964266427456399126032383, 7.880894369283524032753266031237, 8.482736968977321125413003511592, 9.47735008211383812698647076507, 10.03089703576269715754426333941, 10.789362918466628926518630034816, 11.486445455708696776900743564592, 12.22386128547445858771851034264, 12.94806360131784438827040527599, 13.58710402689122900644429885394, 14.55130012457338831078009874147, 15.362300046065336891339800992584, 16.36253547549256977428491682953, 16.83980649863756094828708917187, 17.485311701635920763947173103293, 18.424240410997833077836814869489, 18.92072782740082148226541717411, 19.48603700511868579342757003138

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