Properties

Label 1-2205-2205.2084-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.965 - 0.260i$
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  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)8-s + (0.988 + 0.149i)11-s + (−0.988 − 0.149i)13-s + (−0.900 + 0.433i)16-s + (0.733 + 0.680i)17-s + (0.5 + 0.866i)19-s + (0.733 − 0.680i)22-s + (0.955 + 0.294i)23-s + (−0.733 + 0.680i)26-s + (0.733 + 0.680i)29-s − 31-s + (−0.222 + 0.974i)32-s + (0.988 − 0.149i)34-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)8-s + (0.988 + 0.149i)11-s + (−0.988 − 0.149i)13-s + (−0.900 + 0.433i)16-s + (0.733 + 0.680i)17-s + (0.5 + 0.866i)19-s + (0.733 − 0.680i)22-s + (0.955 + 0.294i)23-s + (−0.733 + 0.680i)26-s + (0.733 + 0.680i)29-s − 31-s + (−0.222 + 0.974i)32-s + (0.988 − 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.965 - 0.260i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.965 - 0.260i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.042730296 - 0.2707812889i\)
\(L(\frac12)\) \(\approx\) \(2.042730296 - 0.2707812889i\)
\(L(1)\) \(\approx\) \(1.331820785 - 0.4655950090i\)
\(L(1)\) \(\approx\) \(1.331820785 - 0.4655950090i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.955 + 0.294i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.955 + 0.294i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−19.73607658830725398560107111964, −19.08465397506814086821543184209, −18.08478540573805453592222723966, −17.40654459224075920392691944433, −16.76272611260869799198794633843, −16.234433643262526167610481980361, −15.24801628971252327686890474350, −14.77326301298793995818584870316, −13.95126736837089866543219964634, −13.52488627888387070382577772737, −12.376150199065804944676224731208, −12.01576831822262468890082104756, −11.22986727436432297742902287707, −10.07029496412087776006918020243, −9.12586243922961240666101996839, −8.72962852796413673054797468017, −7.41743766374683270710247915555, −7.16964960686959377850440470045, −6.29588257709761172085423267190, −5.286583148167460519821410798451, −4.82964614799852454196751265102, −3.80561029177547324488494969103, −3.079182569046282266330476631062, −2.119826314574011747686816392418, −0.59756089090680855039595857883, 1.15566514883914779883085532824, 1.7246440465543992975176026400, 2.97182183955731210662834424614, 3.5032646946170365308472760007, 4.47842364654545628606960671799, 5.18643026568768731293462274104, 6.00472803514240133795169900839, 6.83044781919610876044636307607, 7.739432008531489999605750707060, 8.84917123907447483891265874451, 9.57967735526486086676900109584, 10.18470992142309662594650409638, 10.98458374451672559222846346638, 11.85026470702293929786488368935, 12.34092952958153497315695876480, 12.974680763200105118878511286752, 13.94345295352648952968009552717, 14.70954442603646186201147827228, 14.843192323019017413362324384, 16.06974956386806670283539499235, 16.8564646393085838205343653735, 17.620415112204112594872882411795, 18.462552933519545235785623462400, 19.25876619632612657970302267621, 19.71072938111106842341722014444

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