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Properties

Label 2-1881-209.208-c1-0-30
Degree $2$
Conductor $1881$
Sign $-0.216 - 0.976i$
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  + 2.29·2-s + 3.24·4-s − 2.24·5-s + 2.78i·7-s + 2.85·8-s − 5.14·10-s + (3.31 + 0.0382i)11-s − 6.74·13-s + 6.38i·14-s + 0.0447·16-s + 6.10i·17-s + (0.994 + 4.24i)19-s − 7.29·20-s + (7.59 + 0.0876i)22-s + 5.56·23-s + ⋯
L(s)  = 1  + 1.61·2-s + 1.62·4-s − 1.00·5-s + 1.05i·7-s + 1.00·8-s − 1.62·10-s + (0.999 + 0.0115i)11-s − 1.87·13-s + 1.70i·14-s + 0.0111·16-s + 1.48i·17-s + (0.228 + 0.973i)19-s − 1.63·20-s + (1.61 + 0.0186i)22-s + 1.16·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1881\)    =    \(3^{2} \cdot 11 \cdot 19\)
Sign: $-0.216 - 0.976i$
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: \((2,\ 1881,\ (\ :1/2),\ -0.216 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.812927257\)
\(L(\frac12)\) \(\approx\) \(2.812927257\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 0.0382i)T \)
19 \( 1 + (-0.994 - 4.24i)T \)
good2 \( 1 - 2.29T + 2T^{2} \)
5 \( 1 + 2.24T + 5T^{2} \)
7 \( 1 - 2.78iT - 7T^{2} \)
13 \( 1 + 6.74T + 13T^{2} \)
17 \( 1 - 6.10iT - 17T^{2} \)
23 \( 1 - 5.56T + 23T^{2} \)
29 \( 1 + 3.42T + 29T^{2} \)
31 \( 1 - 6.66iT - 31T^{2} \)
37 \( 1 + 1.10iT - 37T^{2} \)
41 \( 1 - 4.69T + 41T^{2} \)
43 \( 1 + 7.61iT - 43T^{2} \)
47 \( 1 - 5.34T + 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 - 9.96iT - 61T^{2} \)
67 \( 1 + 6.39iT - 67T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 - 2.27iT - 73T^{2} \)
79 \( 1 + 7.84T + 79T^{2} \)
83 \( 1 + 15.4iT - 83T^{2} \)
89 \( 1 - 4.77iT - 89T^{2} \)
97 \( 1 - 14.8iT - 97T^{2} \)
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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

−9.338570018565173716721893306039, −8.637830392969651921416778116439, −7.55897327737333145457648156170, −6.95847100839101622319797650837, −5.95699861637794880488663142397, −5.35943197305851320757790556895, −4.43707174649335655853330833488, −3.79208494579440500767906471716, −2.93133549936345328858420476578, −1.88858575271740291916988706339, 0.56283687687746734952784776242, 2.48573000695453179265730838843, 3.29776208353696691874090248579, 4.27963906991284586426176836637, 4.60579829962267019289218339430, 5.45929856556949644553062692730, 6.77944398943514017472920401325, 7.18748349162981386708803482966, 7.66831796538312818373576794748, 9.178536055717067783936867203943

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