↑

Properties

Label 2-1098-183.101-c1-0-2
Degree $2$
Conductor $1098$
Sign $-0.772 - 0.634i$
Analytic cond. $8.76757$
Root an. cond. $2.96100$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.69 − 2.93i)5-s + (0.229 + 0.857i)7-s + (−0.707 + 0.707i)8-s + (−0.878 + 3.27i)10-s + (−4.13 + 4.13i)11-s + (−1.60 + 2.77i)13-s + (−0.443 − 0.768i)14-s + (0.500 − 0.866i)16-s + (−2.21 − 0.592i)17-s + (−6.05 + 3.49i)19-s − 3.39i·20-s + (2.92 − 5.07i)22-s + (−6.28 − 6.28i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (0.758 − 1.31i)5-s + (0.0868 + 0.324i)7-s + (−0.249 + 0.249i)8-s + (−0.277 + 1.03i)10-s + (−1.24 + 1.24i)11-s + (−0.444 + 0.769i)13-s + (−0.118 − 0.205i)14-s + (0.125 − 0.216i)16-s + (−0.536 − 0.143i)17-s + (−1.38 + 0.802i)19-s − 0.758i·20-s + (0.624 − 1.08i)22-s + (−1.31 − 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1098\)    =    \(2 \cdot 3^{2} \cdot 61\)
Sign: $-0.772 - 0.634i$
Analytic conductor: \(8.76757\)
Root analytic conductor: \(2.96100\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1098} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1098,\ (\ :1/2),\ -0.772 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3375387055\)
\(L(\frac12)\) \(\approx\) \(0.3375387055\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
61 \( 1 + (6.52 + 4.29i)T \)
good5 \( 1 + (-1.69 + 2.93i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.229 - 0.857i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (4.13 - 4.13i)T - 11iT^{2} \)
13 \( 1 + (1.60 - 2.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.21 + 0.592i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.05 - 3.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.28 + 6.28i)T + 23iT^{2} \)
29 \( 1 + (-0.737 - 0.197i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (2.79 - 10.4i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.70 - 5.70i)T + 37iT^{2} \)
41 \( 1 - 0.414T + 41T^{2} \)
43 \( 1 + (5.78 - 1.55i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.33 + 1.34i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.97 - 3.97i)T + 53iT^{2} \)
59 \( 1 + (1.53 + 5.72i)T + (-51.0 + 29.5i)T^{2} \)
67 \( 1 + (-3.62 + 0.970i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.13 + 0.572i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.946 + 1.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.48 + 13.0i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (3.05 + 1.76i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (10.7 - 10.7i)T - 89iT^{2} \)
97 \( 1 + (-12.7 + 7.38i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.15529475914979778310564087641, −9.263838767695691920064556254463, −8.583075348689479189098126587734, −7.994991718582389989133376868743, −6.85399981513952028159352934124, −6.02052749693479341308457769664, −4.96011050431043910696321574117, −4.47024308533655477428824363575, −2.33577935029037892934094221257, −1.75997687535072148324420741627, 0.16595184278699914971705789770, 2.23364463579423133668569622820, 2.77966845114271660128937068336, 3.98481231853472073459092071796, 5.67294036744016437223936651250, 6.09556982616745824605575995515, 7.23696883892103829508723781813, 7.82894965041484630300192075849, 8.716194553185643126069778687336, 9.816527386345704693438536221327

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