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Properties

Label 2-33e2-9.7-c1-0-88
Degree $2$
Conductor $1089$
Sign $-0.861 + 0.506i$
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  + (0.947 − 1.64i)2-s + (0.157 + 1.72i)3-s + (−0.794 − 1.37i)4-s + (−1.75 − 3.04i)5-s + (2.97 + 1.37i)6-s + (1.71 − 2.97i)7-s + 0.777·8-s + (−2.95 + 0.544i)9-s − 6.65·10-s + (2.24 − 1.58i)12-s + (0.634 + 1.09i)13-s + (−3.25 − 5.63i)14-s + (4.97 − 3.50i)15-s + (2.32 − 4.02i)16-s + 0.406·17-s + (−1.90 + 5.35i)18-s + ⋯
L(s)  = 1  + (0.669 − 1.16i)2-s + (0.0911 + 0.995i)3-s + (−0.397 − 0.688i)4-s + (−0.785 − 1.36i)5-s + (1.21 + 0.561i)6-s + (0.648 − 1.12i)7-s + 0.274·8-s + (−0.983 + 0.181i)9-s − 2.10·10-s + (0.649 − 0.458i)12-s + (0.175 + 0.304i)13-s + (−0.868 − 1.50i)14-s + (1.28 − 0.906i)15-s + (0.581 − 1.00i)16-s + 0.0985·17-s + (−0.448 + 1.26i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.861 + 0.506i$
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: \((2,\ 1089,\ (\ :1/2),\ -0.861 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.914813234\)
\(L(\frac12)\) \(\approx\) \(1.914813234\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.157 - 1.72i)T \)
11 \( 1 \)
good2 \( 1 + (-0.947 + 1.64i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.75 + 3.04i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.71 + 2.97i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-0.634 - 1.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.406T + 17T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 + (2.54 + 4.41i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.275 - 0.477i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.443 - 0.767i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.26T + 37T^{2} \)
41 \( 1 + (1.16 + 2.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.79 + 3.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.99 + 8.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + (3.60 + 6.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.769 - 1.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.84 - 4.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.59T + 71T^{2} \)
73 \( 1 + 1.43T + 73T^{2} \)
79 \( 1 + (6.60 - 11.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.764 + 1.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.09T + 89T^{2} \)
97 \( 1 + (1.72 - 2.98i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.847469883424679648237238475720, −8.648615794661388177859826882554, −8.286137119458626486811718835276, −7.15874451974549694935639594730, −5.48842927710367217532765062256, −4.59807550608869803130867899342, −4.21360184424194563147125887901, −3.64488787628067658539105524832, −2.09259403925100745667957614501, −0.67187886781875456689069726805, 1.95613705570883875622568959612, 3.03851270505834955610148791672, 4.19485950296204754269897677153, 5.49876511234502643550805243325, 6.10159838912272948193084424792, 6.81256132434620865571730840488, 7.61889433791825962783180281764, 8.051364611037984414086001544336, 8.888864778315402729205604860283, 10.44636192017344276368383701043

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