↑

Properties

Label 1-2205-2205.1318-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.196 + 0.980i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (0.781 + 0.623i)8-s + (0.955 − 0.294i)11-s + (0.294 + 0.955i)13-s + (0.623 + 0.781i)16-s + (−0.997 − 0.0747i)17-s + (0.5 + 0.866i)19-s + (0.997 − 0.0747i)22-s + (0.563 + 0.826i)23-s + (0.0747 + 0.997i)26-s + (−0.0747 + 0.997i)29-s + 31-s + (0.433 + 0.900i)32-s + (−0.955 − 0.294i)34-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (0.781 + 0.623i)8-s + (0.955 − 0.294i)11-s + (0.294 + 0.955i)13-s + (0.623 + 0.781i)16-s + (−0.997 − 0.0747i)17-s + (0.5 + 0.866i)19-s + (0.997 − 0.0747i)22-s + (0.563 + 0.826i)23-s + (0.0747 + 0.997i)26-s + (−0.0747 + 0.997i)29-s + 31-s + (0.433 + 0.900i)32-s + (−0.955 − 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.196 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.196138430 + 3.901009648i\)
\(L(\frac12)\) \(\approx\) \(3.196138430 + 3.901009648i\)
\(L(1)\) \(\approx\) \(2.028468225 + 0.7404983563i\)
\(L(1)\) \(\approx\) \(2.028468225 + 0.7404983563i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.974 + 0.222i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (0.294 + 0.955i)T \)
17 \( 1 + (-0.997 - 0.0747i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.563 + 0.826i)T \)
29 \( 1 + (-0.0747 + 0.997i)T \)
31 \( 1 + T \)
37 \( 1 + (0.563 - 0.826i)T \)
41 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (-0.149 - 0.988i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (0.563 + 0.826i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.294 + 0.955i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.294 + 0.955i)T \)
89 \( 1 + (0.733 - 0.680i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.61222994181511736422696695084, −18.83081118294170525600743300559, −17.83306608328330836313206976837, −17.14714301130639336617368357573, −16.33226871936960186178041451488, −15.28098765242071540023365954497, −15.21750926892165824736112556508, −14.2045407808869684965544735516, −13.34804110276051391797628782889, −13.05186830013705873649337549107, −11.93919866840189305165141303820, −11.548646248301746289047053975262, −10.66709646283687396692414096935, −9.9740555085959835976580770693, −9.04026740837402983629303048420, −8.11792473872145976895820385777, −7.093075119178957864992554790538, −6.48543906513003285464741358503, −5.77476741828309398750928973561, −4.68694194621998211956885559228, −4.30639808381185629893849807706, −3.1619816722414832343443225100, −2.57548227788904336912717353389, −1.455843391327861587032267309395, −0.57123465066955499122873688351, 1.21058094778623342048818835986, 1.95657258421254375543247319812, 3.071376289720854536876687480828, 3.851280077540352056719117003419, 4.47194507096469592687978599797, 5.43381127979068155325774285442, 6.221772397594345074121371183204, 6.87037579784008328288475352671, 7.56252676625309160094133589575, 8.68017484354589048456977227766, 9.2385421675791988080183591110, 10.43636026548284079404127101807, 11.22384744804841859202031379574, 11.82050521498535136750280363190, 12.41977408220825616521602449911, 13.47843848500520098880987581386, 13.88711519703246548973080319625, 14.55411645998379989801559082001, 15.39957769886242579626171354676, 16.00932014088679261377560231184, 16.82012736667936243711133814079, 17.26057267108401347763229026644, 18.370366697965071425656186200766, 19.17743194848915298135269028480, 19.95329965008788625735850601626

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