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Properties

Label 1-2205-2205.562-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.958 - 0.285i$
Analytic cond. $236.960$
Root an. cond. $236.960$
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  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (−0.988 − 0.149i)11-s + (−0.149 + 0.988i)13-s + (−0.900 + 0.433i)16-s + (0.680 − 0.733i)17-s + (0.5 + 0.866i)19-s + (−0.680 − 0.733i)22-s + (0.294 − 0.955i)23-s + (−0.733 + 0.680i)26-s + (0.733 + 0.680i)29-s + 31-s + (−0.974 − 0.222i)32-s + (0.988 − 0.149i)34-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (−0.988 − 0.149i)11-s + (−0.149 + 0.988i)13-s + (−0.900 + 0.433i)16-s + (0.680 − 0.733i)17-s + (0.5 + 0.866i)19-s + (−0.680 − 0.733i)22-s + (0.294 − 0.955i)23-s + (−0.733 + 0.680i)26-s + (0.733 + 0.680i)29-s + 31-s + (−0.974 − 0.222i)32-s + (0.988 − 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.958 - 0.285i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.958 - 0.285i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3227657998 + 2.210394979i\)
\(L(\frac12)\) \(\approx\) \(-0.3227657998 + 2.210394979i\)
\(L(1)\) \(\approx\) \(1.197847672 + 0.8403810366i\)
\(L(1)\) \(\approx\) \(1.197847672 + 0.8403810366i\)

Euler product

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

−19.36442277753347899295991606670, −18.59255671986775064102576243253, −17.81611291848878137626196423472, −17.13154402001615983991906469044, −15.84015971669288041892005134103, −15.4963734942802423941215738382, −14.839443183393686993775715675828, −13.828046743539617125340805152470, −13.33763569559840156144601268921, −12.61732813602148715697683811828, −11.96761894228775578138727637180, −11.13231249827143043516307466520, −10.333189266106454315195901537300, −9.95657922749890789872448853279, −8.89906074762861858969706509893, −7.864838555164069272658339366053, −7.17371001475664847053627693525, −6.04708068613585287738148315946, −5.421485595674644314516262348169, −4.78893883722240562013060187553, −3.75819552200104335603867425014, −2.96576825870959908011577270259, −2.316787727663059251350266052436, −1.14691115351053331361839635623, −0.29257614145230552110925677183, 1.19581477975109001753842522000, 2.54439094350491920860162876325, 3.05364198134192784595680792879, 4.1507595457421140322374850774, 4.89601968552119046359555301036, 5.52009558208036485214098037556, 6.49793146293080484470353917949, 7.07738059313415297519539486052, 8.026345864345136859529149953697, 8.4921752577620964472218907789, 9.63206224642101990440343481979, 10.39476518092974307903198233303, 11.48585625038674627041773007386, 11.99452290766305763886580519530, 12.76917616646846986771944308074, 13.56165603690948169468738894205, 14.16523230143490648751457052287, 14.77401943973898892578604708191, 15.64083915950557067758146444039, 16.468373688184470036318765715905, 16.57145692667015793549659939923, 17.76024450313003192069799300168, 18.42511798794816682381613592067, 19.07758963399609766076422470259, 20.26232237953867983945092751791

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