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Properties

Label 2-33e2-9.4-c1-0-19
Degree $2$
Conductor $1089$
Sign $0.508 - 0.861i$
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  + (−0.288 − 0.500i)2-s + (−1.33 + 1.10i)3-s + (0.833 − 1.44i)4-s + (−1.50 + 2.60i)5-s + (0.938 + 0.346i)6-s + (0.582 + 1.00i)7-s − 2.11·8-s + (0.549 − 2.94i)9-s + 1.73·10-s + (0.487 + 2.84i)12-s + (1.97 − 3.41i)13-s + (0.336 − 0.582i)14-s + (−0.878 − 5.12i)15-s + (−1.05 − 1.82i)16-s − 0.314·17-s + (−1.63 + 0.577i)18-s + ⋯
L(s)  = 1  + (−0.204 − 0.353i)2-s + (−0.769 + 0.639i)3-s + (0.416 − 0.721i)4-s + (−0.671 + 1.16i)5-s + (0.383 + 0.141i)6-s + (0.220 + 0.381i)7-s − 0.748·8-s + (0.183 − 0.983i)9-s + 0.548·10-s + (0.140 + 0.821i)12-s + (0.547 − 0.947i)13-s + (0.0899 − 0.155i)14-s + (−0.226 − 1.32i)15-s + (−0.263 − 0.456i)16-s − 0.0762·17-s + (−0.385 + 0.136i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.508 - 0.861i$
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: \((2,\ 1089,\ (\ :1/2),\ 0.508 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9641802364\)
\(L(\frac12)\) \(\approx\) \(0.9641802364\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.33 - 1.10i)T \)
11 \( 1 \)
good2 \( 1 + (0.288 + 0.500i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.50 - 2.60i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.582 - 1.00i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-1.97 + 3.41i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.314T + 17T^{2} \)
19 \( 1 - 6.36T + 19T^{2} \)
23 \( 1 + (-0.0427 + 0.0740i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.84 - 6.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.26 - 5.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 + (2.90 - 5.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.39 - 5.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.163 - 0.283i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.42T + 53T^{2} \)
59 \( 1 + (-1.14 + 1.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.84 - 11.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.83 - 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.71T + 71T^{2} \)
73 \( 1 - 2.94T + 73T^{2} \)
79 \( 1 + (-4.44 - 7.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.47 + 4.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.12T + 89T^{2} \)
97 \( 1 + (-0.0444 - 0.0769i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.46070837778948259156779695460, −9.472036688780818165254655941668, −8.519643630197384353291366544377, −7.24543955927578712060083337483, −6.64738393485729812832046240792, −5.65018345756361644270578606771, −5.06064949418891510277575446864, −3.51208779253188100929964814176, −2.93175045967315076689724228726, −1.13461213007174326011894548411, 0.60407530899053745942241170441, 1.97037779087935003027174073809, 3.67073554450258526298331970664, 4.55591245431578861899967836765, 5.55056772768174829322845757408, 6.52979318851746266399536738838, 7.36221192123164025276498951116, 7.910114838743554738023547860526, 8.643957989135771915031405984268, 9.505349260113856036573040905938

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