↑

Properties

Label 1-2205-2205.1987-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.0590 + 0.998i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.149 − 0.988i)2-s + (−0.955 − 0.294i)4-s + (−0.433 + 0.900i)8-s + (−0.988 − 0.149i)11-s + (−0.930 + 0.365i)13-s + (0.826 + 0.563i)16-s + (0.974 + 0.222i)17-s + 19-s + (−0.294 + 0.955i)22-s + (0.294 − 0.955i)23-s + (0.222 + 0.974i)26-s + (−0.955 + 0.294i)29-s + (0.5 − 0.866i)31-s + (0.680 − 0.733i)32-s + (0.365 − 0.930i)34-s + ⋯
L(s)  = 1  + (0.149 − 0.988i)2-s + (−0.955 − 0.294i)4-s + (−0.433 + 0.900i)8-s + (−0.988 − 0.149i)11-s + (−0.930 + 0.365i)13-s + (0.826 + 0.563i)16-s + (0.974 + 0.222i)17-s + 19-s + (−0.294 + 0.955i)22-s + (0.294 − 0.955i)23-s + (0.222 + 0.974i)26-s + (−0.955 + 0.294i)29-s + (0.5 − 0.866i)31-s + (0.680 − 0.733i)32-s + (0.365 − 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0590 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0590 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.0590 + 0.998i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1987, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.0590 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1182561383 + 0.1114690220i\)
\(L(\frac12)\) \(\approx\) \(0.1182561383 + 0.1114690220i\)
\(L(1)\) \(\approx\) \(0.7024496675 - 0.3606978495i\)
\(L(1)\) \(\approx\) \(0.7024496675 - 0.3606978495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.149 - 0.988i)T \)
11 \( 1 + (-0.988 - 0.149i)T \)
13 \( 1 + (-0.930 + 0.365i)T \)
17 \( 1 + (0.974 + 0.222i)T \)
19 \( 1 + T \)
23 \( 1 + (0.294 - 0.955i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (0.563 - 0.826i)T \)
47 \( 1 + (-0.149 + 0.988i)T \)
53 \( 1 + (-0.974 + 0.222i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.781 + 0.623i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.930 + 0.365i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
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

−19.30053566271019309493352505399, −18.74565449632637153414539488064, −17.874178433752945555541208754594, −17.43489436412691919716836779056, −16.56269662562191556152107905522, −15.92991343736410543854073200635, −15.253183575228555830058007433214, −14.64046313123149228809731894626, −13.776476884069268394615738229275, −13.25212027226440357787124016123, −12.34867868760762830532628647770, −11.79507181485817743516227099963, −10.46787075605106981926206516377, −9.86181303151523486360725574835, −9.16369220765476505545490662234, −8.15115028729076915630731910425, −7.49059119919650340050099831156, −7.103495390844021341048644996937, −5.83274937845459438958354487517, −5.28226301982057477669024198444, −4.72702746663329858235629978577, −3.4585886997156371556603668802, −2.90325391832951924504595305184, −1.43817717109377109244573496124, −0.055110654064933306818039864721, 1.177078185800162001992255267254, 2.20643031892971700834907288510, 2.94057208889465601985255157554, 3.70431873078103289355708387984, 4.81486101359432847464673049828, 5.24997290198348264753524009662, 6.19857532708295451498182836023, 7.47934732106627090015048857742, 8.02297889874064341689512375531, 9.08213335156873497616804700358, 9.71888293297900056234823995305, 10.41972383739047312087441572460, 11.06773510394321731298023787438, 12.01375503244597601156617064203, 12.444405680470100393922224711483, 13.268064771643416420479047045257, 13.99421768813765181174709893037, 14.645193501949673518098905915006, 15.41702908519770130493647561729, 16.45534869913293866598673159484, 17.12412462934524377481505732589, 17.95519149430172600403889806782, 18.79654102985342281487983967107, 19.00318803291210320500358068498, 20.01565162294718669307289494328

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