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Properties

Label 1-2205-2205.988-r1-0-0
Degree $1$
Conductor $2205$
Sign $0.931 + 0.363i$
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  + (−0.294 − 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

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

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.931 + 0.363i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (988, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ 0.931 + 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.527770918 + 0.2878986670i\)
\(L(\frac12)\) \(\approx\) \(1.527770918 + 0.2878986670i\)
\(L(1)\) \(\approx\) \(0.8840048336 - 0.2792649583i\)
\(L(1)\) \(\approx\) \(0.8840048336 - 0.2792649583i\)

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

−19.10192994087048117850699791964, −18.84596541845341424142147596273, −17.97801541102405066084465912160, −17.20741078536034898011197537472, −16.595579991792798936132493868266, −16.085858572102055806217431821, −15.14809690150801504010172695346, −14.45972142942167787098138998831, −14.04966404880911808841002821030, −13.051843332215583863765355334763, −12.39960827839785230989902381183, −11.26959352721996975890288423717, −10.68730262798634639901490207431, −9.505744121501018457400240739869, −9.15983174657911964671942989344, −8.42364739699098124180688607240, −7.39852974862506336541490804804, −6.85545688428305001460548674344, −6.09953274446785192721606352006, −5.32987225108420006996388841940, −4.25971804714227989751159900750, −3.86199574574499479013939371708, −2.34485652939391367331954910293, −1.30164460815421585902878476694, −0.36828864132586981705544550077, 0.90634413283381867509673182387, 1.525050833715913409303253418007, 2.57408033141891616315069963039, 3.640163355531143671212717226285, 3.92409123005776785113321999903, 5.13964768046842369655533135953, 5.965440362428442343019469879643, 6.93130377159128627534410665962, 8.06693988492168282998842414938, 8.500304082400029466573422253756, 9.359566194138604351893828856657, 10.0668930133067681717768662131, 10.91720047856723338708273642012, 11.37311611303480756344800338324, 12.24843048626742950012452395668, 13.03030744860070918380989932847, 13.44179437740041086190069499792, 14.55713901913479746316590408009, 15.020362913912235526940595455965, 16.2428054935043356788774569630, 16.92702941448145287940174147684, 17.5331309968442120879479171424, 18.21003464239235772223991648530, 19.20638922607140603449882479909, 19.39913743643383868291116152813

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