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Properties

Label 1-2205-2205.2048-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.419 + 0.907i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.419 + 0.907i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5296052118 + 0.3388353906i\)
\(L(\frac12)\) \(\approx\) \(0.5296052118 + 0.3388353906i\)
\(L(1)\) \(\approx\) \(0.6632104866 - 0.1000280096i\)
\(L(1)\) \(\approx\) \(0.6632104866 - 0.1000280096i\)

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

−19.41745311526996781611784286101, −18.697979375502935510042661791825, −18.308746886489977782047344003296, −17.17894553324994755763763695789, −16.84801236033909960270587702492, −16.085206186083213428308753718430, −15.446398001050835089623977963258, −14.50594403579705621404959413799, −14.07254532788956219506123880442, −13.29952155085035085258701528559, −12.193018839974202118263801566081, −11.37348163475846905977743924181, −10.68076623124920802271981998330, −9.87801865661402182892286175563, −9.08857957137408571399331914800, −8.51397030851331346385025162815, −7.751806763970361067844381417255, −6.74536508511463605791428376062, −6.40355267677438473068347428312, −5.32378959914221244082671855295, −4.67191655908743259767534934406, −3.59867024827145870008967293179, −2.392579983030224655495309805663, −1.48816508381388622700233478880, −0.30106613478257664909770476884, 1.02014608793946242835827299064, 2.03458740075412121234556218957, 2.754326379284628305677913881555, 3.65559377232455494902323593102, 4.62548571956062472625270131488, 5.28599098046442628255864808255, 6.86061376312157098570617656644, 7.11748301295589730598544782924, 8.15374625960268684406498600064, 8.85529026933237482813481028780, 9.72622912393650240836868837415, 10.116392895236513744766368166896, 11.12636044757475101330608453174, 11.67683701533277050373521364698, 12.51385779233937964416974961205, 13.09507957326041359711044519712, 13.83390560861223034495904711736, 15.00637329556008564844918301982, 15.529329528455219344667857606637, 16.42880516152554897592416164183, 17.32233531602386191140058355334, 17.813659331538278442280484124823, 18.18656391919253668582023587853, 19.56875384623800825655312604263, 19.6813929993601176138990962923

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