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Properties

Label 2-33e2-9.7-c1-0-23
Degree $2$
Conductor $1089$
Sign $0.953 - 0.302i$
Analytic cond. $8.69570$
Root an. cond. $2.94884$
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  + (0.142 − 0.246i)2-s + (0.0364 − 1.73i)3-s + (0.959 + 1.66i)4-s + (−1.35 − 2.34i)5-s + (−0.422 − 0.255i)6-s + (−2.03 + 3.52i)7-s + 1.11·8-s + (−2.99 − 0.126i)9-s − 0.772·10-s + (2.91 − 1.60i)12-s + (1.92 + 3.33i)13-s + (0.580 + 1.00i)14-s + (−4.11 + 2.25i)15-s + (−1.75 + 3.04i)16-s + 4.32·17-s + (−0.458 + 0.722i)18-s + ⋯
L(s)  = 1  + (0.100 − 0.174i)2-s + (0.0210 − 0.999i)3-s + (0.479 + 0.830i)4-s + (−0.605 − 1.04i)5-s + (−0.172 − 0.104i)6-s + (−0.769 + 1.33i)7-s + 0.395·8-s + (−0.999 − 0.0421i)9-s − 0.244·10-s + (0.840 − 0.462i)12-s + (0.534 + 0.924i)13-s + (0.155 + 0.268i)14-s + (−1.06 + 0.583i)15-s + (−0.439 + 0.761i)16-s + 1.04·17-s + (−0.108 + 0.170i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.953 - 0.302i$
Analytic conductor: \(8.69570\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1/2),\ 0.953 - 0.302i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.482036106\)
\(L(\frac12)\) \(\approx\) \(1.482036106\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0364 + 1.73i)T \)
11 \( 1 \)
good2 \( 1 + (-0.142 + 0.246i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.35 + 2.34i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.03 - 3.52i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-1.92 - 3.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 - 1.62T + 19T^{2} \)
23 \( 1 + (-0.932 - 1.61i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.77 + 3.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.43 - 4.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.74T + 37T^{2} \)
41 \( 1 + (-3.43 - 5.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.492 + 0.853i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.93 - 5.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.57T + 53T^{2} \)
59 \( 1 + (5.68 + 9.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.30 - 7.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.870 - 1.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.74T + 71T^{2} \)
73 \( 1 - 4.04T + 73T^{2} \)
79 \( 1 + (2.14 - 3.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.25 - 5.62i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.26T + 89T^{2} \)
97 \( 1 + (-3.70 + 6.41i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.592584338548952746375136062014, −8.859799828596976428773769020682, −8.219176945437342678415130528736, −7.59551222811200073822976606959, −6.51272288859730395114213582176, −5.87867641193224545680579123829, −4.66276866074788865558726006329, −3.40052122674708926572647978871, −2.58980648298472202580454874444, −1.30343358311315182654230939991, 0.70330241831311694076527137077, 2.92035617350644230486377782962, 3.50688312669205344366443564798, 4.49622380678850860782590129293, 5.67150964403654310886369759174, 6.38989056150428985319461070075, 7.30365144479810538385219885770, 7.919936613987419700591980151853, 9.354109795980738751209990595742, 10.16049257629523260617015619127

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