↑

Properties

Label 2-1098-9.7-c1-0-1
Degree $2$
Conductor $1098$
Sign $0.144 - 0.989i$
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  + (−0.5 + 0.866i)2-s + (−0.887 − 1.48i)3-s + (−0.499 − 0.866i)4-s + (−2.04 − 3.55i)5-s + (1.73 − 0.0253i)6-s + (−1.01 + 1.76i)7-s + 0.999·8-s + (−1.42 + 2.64i)9-s + 4.09·10-s + (−0.373 + 0.646i)11-s + (−0.843 + 1.51i)12-s + (−1.26 − 2.19i)13-s + (−1.01 − 1.76i)14-s + (−3.45 + 6.20i)15-s + (−0.5 + 0.866i)16-s − 6.94·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.512 − 0.858i)3-s + (−0.249 − 0.433i)4-s + (−0.916 − 1.58i)5-s + (0.707 − 0.0103i)6-s + (−0.384 + 0.666i)7-s + 0.353·8-s + (−0.474 + 0.880i)9-s + 1.29·10-s + (−0.112 + 0.194i)11-s + (−0.243 + 0.436i)12-s + (−0.351 − 0.609i)13-s + (−0.272 − 0.471i)14-s + (−0.893 + 1.60i)15-s + (−0.125 + 0.216i)16-s − 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1098\)    =    \(2 \cdot 3^{2} \cdot 61\)
Sign: $0.144 - 0.989i$
Analytic conductor: \(8.76757\)
Root analytic conductor: \(2.96100\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1098} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1098,\ (\ :1/2),\ 0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2132078406\)
\(L(\frac12)\) \(\approx\) \(0.2132078406\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.887 + 1.48i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (2.04 + 3.55i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.01 - 1.76i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.373 - 0.646i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.26 + 2.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.94T + 17T^{2} \)
19 \( 1 - 4.45T + 19T^{2} \)
23 \( 1 + (2.94 + 5.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.87 + 3.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.13 + 1.96i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.890T + 37T^{2} \)
41 \( 1 + (-4.15 - 7.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.79 - 6.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.632 - 1.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.98T + 53T^{2} \)
59 \( 1 + (-4.87 - 8.44i)T + (-29.5 + 51.0i)T^{2} \)
67 \( 1 + (-5.67 - 9.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.97T + 71T^{2} \)
73 \( 1 - 9.19T + 73T^{2} \)
79 \( 1 + (-3.16 + 5.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.05 + 13.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.90T + 89T^{2} \)
97 \( 1 + (6.30 - 10.9i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.741455557703674044662304782591, −8.961548620508075672288469939093, −8.208604946726224127667574676683, −7.77862401180558241803470225571, −6.73415903618349832685863930831, −5.85658108841534857598299595672, −5.02515065312083164348597914862, −4.33014106869550830249580196368, −2.44883805809955476028574788617, −0.929744907315696037268935977356, 0.14790336062001609750858090504, 2.46464805613640135614683297253, 3.66200261073216991152791623068, 3.86993964559070792945056842138, 5.17992597919712886098625372444, 6.66863161203667712358435152949, 6.97945006889652125687623728487, 8.040494023970602127086737057807, 9.180996449959244886758177444950, 9.889443694136875898378855464453

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