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Properties

Label 1-2205-2205.1154-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.818 - 0.575i$
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  + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (−0.900 − 0.433i)8-s + (0.988 + 0.149i)11-s + (0.365 + 0.930i)13-s + (0.826 + 0.563i)16-s + (0.222 − 0.974i)17-s − 19-s + (−0.955 − 0.294i)22-s + (0.955 + 0.294i)23-s + (−0.222 − 0.974i)26-s + (−0.955 + 0.294i)29-s + (0.5 − 0.866i)31-s + (−0.733 − 0.680i)32-s + (−0.365 + 0.930i)34-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (−0.900 − 0.433i)8-s + (0.988 + 0.149i)11-s + (0.365 + 0.930i)13-s + (0.826 + 0.563i)16-s + (0.222 − 0.974i)17-s − 19-s + (−0.955 − 0.294i)22-s + (0.955 + 0.294i)23-s + (−0.222 − 0.974i)26-s + (−0.955 + 0.294i)29-s + (0.5 − 0.866i)31-s + (−0.733 − 0.680i)32-s + (−0.365 + 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.818 - 0.575i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.818 - 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033540333 - 0.3268861348i\)
\(L(\frac12)\) \(\approx\) \(1.033540333 - 0.3268861348i\)
\(L(1)\) \(\approx\) \(0.7734875072 - 0.07398353966i\)
\(L(1)\) \(\approx\) \(0.7734875072 - 0.07398353966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.988 - 0.149i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (0.222 - 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + (0.826 - 0.563i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (0.988 + 0.149i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.365 + 0.930i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−19.68320040814758265952215619931, −19.05038294799214290980005886314, −18.412207367253048698455610512354, −17.51751596107698367338877822662, −16.987855139665652937451245563705, −16.51973563102782791838784264130, −15.33042003882964269151019952631, −15.078543841461941711493116546526, −14.245324673226953734510696338281, −13.10980423124146391596054550008, −12.430172952481969764913167209460, −11.562341297511216372482788473776, −10.79056259109916728344040825726, −10.30519583834650348012415871107, −9.34240009140848916218316494903, −8.66737315664595050943847933035, −8.08665939457456427720394403340, −7.202803709638514864647798692936, −6.29922135817226357912544997392, −5.90169726207612604628276997946, −4.6728305743226171517385813766, −3.57622105750796584626842422856, −2.7671553070422328935415111221, −1.63393416996378078655244488172, −0.91610448423908410453498538108, 0.64228508855543584830287369401, 1.68378367650233959163819822020, 2.40195975011563187715867983218, 3.534861204032648617094051614309, 4.252901488270973128431336848297, 5.49698485952365029659709262942, 6.473813809560702865626680592191, 6.99435170556573506930263767687, 7.77387426675825809846867006516, 8.83889862115546261235535310263, 9.204371892575450888940627286963, 9.89155499443274059252912595680, 11.01971876947930994112279305132, 11.3436291391366640683532470928, 12.16926888569428685863136364085, 12.92786305931848002196272968467, 13.96669037911442010860658092180, 14.719138797070259823374153074799, 15.469997415443047947184968867640, 16.28891088751108119417192283133, 16.957143565251040997947244590601, 17.34993110113164297578029557665, 18.403105060804153110710143978982, 18.8985424328080043067873738613, 19.4899355641366618417705237944

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