↑

Properties

Label 2-1098-183.101-c1-0-20
Degree $2$
Conductor $1098$
Sign $-0.123 + 0.992i$
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 + (2.02 − 3.51i)5-s + (−0.376 − 1.40i)7-s + (0.707 − 0.707i)8-s + (1.04 − 3.91i)10-s + (−2.07 + 2.07i)11-s + (2.10 − 3.64i)13-s + (−0.728 − 1.26i)14-s + (0.500 − 0.866i)16-s + (2.30 + 0.617i)17-s + (−3.11 + 1.80i)19-s − 4.05i·20-s + (−1.46 + 2.54i)22-s + (4.74 + 4.74i)23-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.906 − 1.57i)5-s + (−0.142 − 0.531i)7-s + (0.249 − 0.249i)8-s + (0.331 − 1.23i)10-s + (−0.626 + 0.626i)11-s + (0.584 − 1.01i)13-s + (−0.194 − 0.337i)14-s + (0.125 − 0.216i)16-s + (0.558 + 0.149i)17-s + (−0.715 + 0.413i)19-s − 0.906i·20-s + (−0.313 + 0.542i)22-s + (0.989 + 0.989i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1098\)    =    \(2 \cdot 3^{2} \cdot 61\)
Sign: $-0.123 + 0.992i$
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: \((2,\ 1098,\ (\ :1/2),\ -0.123 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.727788990\)
\(L(\frac12)\) \(\approx\) \(2.727788990\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.31 - 2.73i)T \)
good5 \( 1 + (-2.02 + 3.51i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.376 + 1.40i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.07 - 2.07i)T - 11iT^{2} \)
13 \( 1 + (-2.10 + 3.64i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.30 - 0.617i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.11 - 1.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.74 - 4.74i)T + 23iT^{2} \)
29 \( 1 + (5.85 + 1.56i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.0233 - 0.0871i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.07 - 2.07i)T + 37iT^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + (-4.34 + 1.16i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.19 + 1.26i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.83 + 1.83i)T + 53iT^{2} \)
59 \( 1 + (-3.27 - 12.2i)T + (-51.0 + 29.5i)T^{2} \)
67 \( 1 + (-5.71 + 1.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.88 - 2.38i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.863 + 1.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.68 + 6.27i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (3.26 + 1.88i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (13.3 - 13.3i)T - 89iT^{2} \)
97 \( 1 + (-9.18 + 5.30i)T + (48.5 - 84.0i)T^{2} \)
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

−9.878026682265753194797233696563, −8.829849129226760141618132991572, −8.046421047471175574464115141916, −7.06888913332312251926688887309, −5.74174258261306644267879263219, −5.43974277938961408961019387471, −4.50456221051529759184907493737, −3.49752547946673453837440911268, −2.05404714295386899282687792114, −0.994193829111731350084876661334, 2.06909773723131838195310790798, 2.83747001715280827838800226784, 3.71408318154853550421706034203, 5.13426397045171488736088943837, 5.94984869498114793101197211555, 6.59472486565498946303863426785, 7.19466468984692916001912857251, 8.418973979691715045908159555552, 9.315548559315554857813782665001, 10.24621021929270968741249610776

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