Dirichlet series
| L(s) = 1 | − 2·4-s + 3·5-s + 4·16-s − 6·20-s + 9·23-s + 4·25-s − 5·31-s + 7·37-s + 12·47-s − 7·49-s − 6·53-s + 15·59-s − 8·64-s + 13·67-s + 3·71-s + 12·80-s + 9·89-s − 18·92-s + 17·97-s − 8·100-s − 4·103-s − 21·113-s + 27·115-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s + 16-s − 1.34·20-s + 1.87·23-s + 4/5·25-s − 0.898·31-s + 1.15·37-s + 1.75·47-s − 49-s − 0.824·53-s + 1.95·59-s − 64-s + 1.58·67-s + 0.356·71-s + 1.34·80-s + 0.953·89-s − 1.87·92-s + 1.72·97-s − 4/5·100-s − 0.394·103-s − 1.97·113-s + 2.51·115-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(1089\) = \(3^{2} \cdot 11^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.69570\) |
| Root analytic conductor: | \(2.94884\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 1089,\ (\ :1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.671988493\) |
| \(L(\frac12)\) | \(\approx\) | \(1.671988493\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|
| bad | 3 | \( 1 \) | |
| 11 | \( 1 \) | ||
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad | |
| 7 | \( 1 + p T^{2} \) | 1.7.a | |
| 13 | \( 1 + p T^{2} \) | 1.13.a | |
| 17 | \( 1 + p T^{2} \) | 1.17.a | |
| 19 | \( 1 + p T^{2} \) | 1.19.a | |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj | |
| 29 | \( 1 + p T^{2} \) | 1.29.a | |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f | |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah | |
| 41 | \( 1 + p T^{2} \) | 1.41.a | |
| 43 | \( 1 + p T^{2} \) | 1.43.a | |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am | |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g | |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap | |
| 61 | \( 1 + p T^{2} \) | 1.61.a | |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an | |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad | |
| 73 | \( 1 + p T^{2} \) | 1.73.a | |
| 79 | \( 1 + p T^{2} \) | 1.79.a | |
| 83 | \( 1 + p T^{2} \) | 1.83.a | |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj | |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar | |
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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
−9.638742540829787389319183524300, −9.239645513051175281393608873936, −8.468596719955261371677845250157, −7.37520405553462112512317968814, −6.34583551331242224592183459087, −5.46569305050405475214160980267, −4.87628538913627800967773832068, −3.66923064750984294129055958828, −2.44495639562721598403091577465, −1.06294681607902093882729364169, 1.06294681607902093882729364169, 2.44495639562721598403091577465, 3.66923064750984294129055958828, 4.87628538913627800967773832068, 5.46569305050405475214160980267, 6.34583551331242224592183459087, 7.37520405553462112512317968814, 8.468596719955261371677845250157, 9.239645513051175281393608873936, 9.638742540829787389319183524300