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Properties

Label 2-1881-209.208-c1-0-61
Degree $2$
Conductor $1881$
Sign $0.893 + 0.448i$
Analytic cond. $15.0198$
Root an. cond. $3.87554$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.15·2-s − 0.654·4-s + 1.65·5-s + 0.0488i·7-s − 3.07·8-s + 1.91·10-s + (−1.85 − 2.74i)11-s + 3.48·13-s + 0.0566i·14-s − 2.26·16-s + 4.74i·17-s + (3.79 − 2.13i)19-s − 1.08·20-s + (−2.15 − 3.18i)22-s + 9.20·23-s + ⋯
L(s)  = 1  + 0.820·2-s − 0.327·4-s + 0.740·5-s + 0.0184i·7-s − 1.08·8-s + 0.606·10-s + (−0.559 − 0.828i)11-s + 0.966·13-s + 0.0151i·14-s − 0.565·16-s + 1.15i·17-s + (0.871 − 0.489i)19-s − 0.242·20-s + (−0.458 − 0.679i)22-s + 1.91·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1881\)    =    \(3^{2} \cdot 11 \cdot 19\)
Sign: $0.893 + 0.448i$
Analytic conductor: \(15.0198\)
Root analytic conductor: \(3.87554\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1881} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1881,\ (\ :1/2),\ 0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.563027313\)
\(L(\frac12)\) \(\approx\) \(2.563027313\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (1.85 + 2.74i)T \)
19 \( 1 + (-3.79 + 2.13i)T \)
good2 \( 1 - 1.15T + 2T^{2} \)
5 \( 1 - 1.65T + 5T^{2} \)
7 \( 1 - 0.0488iT - 7T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 - 4.74iT - 17T^{2} \)
23 \( 1 - 9.20T + 23T^{2} \)
29 \( 1 + 1.20T + 29T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + 4.13iT - 37T^{2} \)
41 \( 1 - 8.22T + 41T^{2} \)
43 \( 1 - 2.11iT - 43T^{2} \)
47 \( 1 - 1.20T + 47T^{2} \)
53 \( 1 - 0.192iT - 53T^{2} \)
59 \( 1 - 0.879iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + 5.05iT - 67T^{2} \)
71 \( 1 - 0.916iT - 71T^{2} \)
73 \( 1 - 5.88iT - 73T^{2} \)
79 \( 1 - 1.29T + 79T^{2} \)
83 \( 1 - 5.12iT - 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 9.01iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.186903664641348295929806400177, −8.507431732969086424363816636014, −7.61439338056828284810006078422, −6.38553834370371397286084293210, −5.77801839043902936051159373787, −5.29965625536582735685026580573, −4.17347593530681433461762321034, −3.39635037526339095134303736132, −2.44900352085186138322639539987, −0.880992957738986293661529412816, 1.16711426054461409055723946331, 2.64430738188761869250258511138, 3.39443758811537234228988102202, 4.53761112319491436324291856602, 5.23292481910070671989979385041, 5.77152636507264423280931393879, 6.80641120326945709632520733992, 7.53726220779755925712982135661, 8.758656130370347734630803040661, 9.220937538730009615748864219532

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