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Properties

Label 2-33e2-99.38-c0-0-0
Degree $2$
Conductor $1089$
Sign $-0.151 + 0.988i$
Analytic cond. $0.543481$
Root an. cond. $0.737212$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.360 − 1.69i)5-s + (−0.978 − 0.207i)9-s + (0.499 − 0.866i)12-s + (−1.72 + 0.181i)15-s + (0.669 + 0.743i)16-s + (0.360 − 1.69i)20-s + (−1.82 + 0.813i)25-s + (−0.309 + 0.951i)27-s + (0.669 − 0.743i)31-s + (−0.809 − 0.587i)36-s + (−0.809 + 0.587i)37-s + 1.73i·45-s + (0.704 + 1.58i)47-s + (0.809 − 0.587i)48-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.360 − 1.69i)5-s + (−0.978 − 0.207i)9-s + (0.499 − 0.866i)12-s + (−1.72 + 0.181i)15-s + (0.669 + 0.743i)16-s + (0.360 − 1.69i)20-s + (−1.82 + 0.813i)25-s + (−0.309 + 0.951i)27-s + (0.669 − 0.743i)31-s + (−0.809 − 0.587i)36-s + (−0.809 + 0.587i)37-s + 1.73i·45-s + (0.704 + 1.58i)47-s + (0.809 − 0.587i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.151 + 0.988i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ -0.151 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178559704\)
\(L(\frac12)\) \(\approx\) \(1.178559704\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.104 + 0.994i)T \)
11 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T^{2} \)
5 \( 1 + (0.360 + 1.69i)T + (-0.913 + 0.406i)T^{2} \)
7 \( 1 + (-0.978 + 0.207i)T^{2} \)
13 \( 1 + (-0.104 - 0.994i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.978 - 0.207i)T^{2} \)
31 \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.704 - 1.58i)T + (-0.669 + 0.743i)T^{2} \)
53 \( 1 + (1.64 - 0.535i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.704 + 1.58i)T + (-0.669 - 0.743i)T^{2} \)
61 \( 1 + (-0.104 + 0.994i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.64 - 0.535i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.104 - 0.994i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \)
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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

−9.621628007899242153387114560762, −8.732866919203795829433570102694, −8.085953218210742556771603173719, −7.56457887788090394294667547604, −6.53555945466509934014532738524, −5.72189369305370569249305708745, −4.67914736702576685346001322458, −3.47024417006653571864797481111, −2.19828096379616610277195692977, −1.12357714568535624351872737169, 2.29157425167734457438647947118, 3.08887070303505115743302534076, 3.88904984305880427926496917222, 5.25092539532557450184918084391, 6.16346037823480867905668232596, 6.90971257128013863199535622360, 7.63377732543839139104005855037, 8.705726974554076102402753410422, 9.873958060237264800573376805915, 10.40554290226392708733555438205

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