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Properties

Label 1-2205-2205.1168-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.363 - 0.931i$
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  + (−0.294 − 0.955i)2-s + (−0.826 + 0.563i)4-s + (0.781 + 0.623i)8-s + (0.955 − 0.294i)11-s + (−0.680 + 0.733i)13-s + (0.365 − 0.930i)16-s + (−0.433 − 0.900i)17-s + 19-s + (−0.563 − 0.826i)22-s + (0.563 + 0.826i)23-s + (0.900 + 0.433i)26-s + (−0.826 − 0.563i)29-s + (0.5 − 0.866i)31-s + (−0.997 − 0.0747i)32-s + (−0.733 + 0.680i)34-s + ⋯
L(s)  = 1  + (−0.294 − 0.955i)2-s + (−0.826 + 0.563i)4-s + (0.781 + 0.623i)8-s + (0.955 − 0.294i)11-s + (−0.680 + 0.733i)13-s + (0.365 − 0.930i)16-s + (−0.433 − 0.900i)17-s + 19-s + (−0.563 − 0.826i)22-s + (0.563 + 0.826i)23-s + (0.900 + 0.433i)26-s + (−0.826 − 0.563i)29-s + (0.5 − 0.866i)31-s + (−0.997 − 0.0747i)32-s + (−0.733 + 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.363 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068335292 - 0.7295376287i\)
\(L(\frac12)\) \(\approx\) \(1.068335292 - 0.7295376287i\)
\(L(1)\) \(\approx\) \(0.8372577679 - 0.3705656221i\)
\(L(1)\) \(\approx\) \(0.8372577679 - 0.3705656221i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.294 - 0.955i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (-0.680 + 0.733i)T \)
17 \( 1 + (-0.433 - 0.900i)T \)
19 \( 1 + T \)
23 \( 1 + (0.563 + 0.826i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.433 + 0.900i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (0.930 + 0.365i)T \)
47 \( 1 + (0.294 + 0.955i)T \)
53 \( 1 + (0.433 - 0.900i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.974 + 0.222i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.680 + 0.733i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
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

−19.862124298688444818382381708393, −19.04543218556298308352594927585, −18.23695823499603851136084173381, −17.58248087757324379681721035151, −17.00139864640682920451741725459, −16.368525290030311400197300596119, −15.48503668032409127976145990029, −14.83251651818448491005364780303, −14.38002288669733998844693947683, −13.46095467687672469512983593662, −12.694360931900794512567464646151, −11.98766253431534919921313526525, −10.788998145342693800636222316445, −10.23393636288840476232022157521, −9.252731638862145971897715274389, −8.84689006526656602575593093630, −7.81970858643150210861611334470, −7.22237710301557601423323557475, −6.45208431488972373037302996540, −5.67852700267621291076310041223, −4.85351005464416390469159586950, −4.10231818280602164978085537971, −3.09841966654849140847208404998, −1.77356830731725673906719417332, −0.797567744753792656793341491848, 0.71251557106800083443313793490, 1.655496856466152884428272799752, 2.57239468535159661353936116405, 3.39539581016580617943216568040, 4.26240167459106672893898807861, 4.95073229227088716162147357095, 5.98437576148656280542561017308, 7.13650070751918953184106601930, 7.67891368365331763358947109542, 8.80455060101373261176540965343, 9.46002416478943505393105270933, 9.76492736794320023050404989755, 11.013227023771115584497733205963, 11.60305664978722318152261429420, 11.95570514393133125200509949794, 12.99468082338810282012435172838, 13.769141856599051754523968923870, 14.19581716025894788550756506785, 15.212690177236467863300453597463, 16.18582648776779187314490501110, 17.015839627895889189488541814992, 17.39072251173355706648811102790, 18.36570617010539703196570502587, 18.98862168714211808054599517076, 19.55481245004629679306783010026

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