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Properties

Label 2-33e2-9.4-c1-0-12
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: $\chi_{1089} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1/2),\ -0.953 - 0.302i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9715306352\)
\(L(\frac12)\) \(\approx\) \(0.9715306352\)
\(L(\frac{3}{2})\) 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

−10.34451725819374477680319478337, −9.452348953737090620253170417099, −8.864191239268736603606523404943, −7.86884140306196391104930253937, −6.68444764132170560508483928465, −6.02263391900588762723459772426, −5.00590722947680629967056977631, −4.21728056275784352340018981530, −2.74579195985608177442823521435, −2.20481199971619911352285222030, 0.41629820872458357705347231813, 1.71545076062969371434009468473, 3.09912014971408689621867249488, 4.24054631938747741462236824007, 5.06904716006227629811685146061, 6.45197394785653093416120769699, 7.24265539409318172070732073485, 7.85924910183185334673517017293, 8.282127450973877036273063507432, 9.099515308565978611492633174040

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