Dirichlet series
| L(s) = 1 | − 2·4-s − 3·9-s + 4·16-s − 17-s + 3·19-s + 5·23-s − 5·25-s + 7·29-s + 6·36-s − 9·37-s + 11·47-s − 7·49-s + 13·59-s − 8·64-s + 2·68-s − 4·71-s − 15·73-s − 6·76-s + 9·81-s + 8·83-s − 17·89-s − 10·92-s + 10·100-s + 12·103-s − 19·107-s − 14·116-s + ⋯ |
| L(s) = 1 | − 4-s − 9-s + 16-s − 0.242·17-s + 0.688·19-s + 1.04·23-s − 25-s + 1.29·29-s + 36-s − 1.47·37-s + 1.60·47-s − 49-s + 1.69·59-s − 64-s + 0.242·68-s − 0.474·71-s − 1.75·73-s − 0.688·76-s + 81-s + 0.878·83-s − 1.80·89-s − 1.04·92-s + 100-s + 1.18·103-s − 1.83·107-s − 1.29·116-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4489 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4489 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(4489\) = \(67^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(35.8448\) |
| Root analytic conductor: | \(5.98705\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 4489,\ (\ :1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | 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 | 67 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + p T^{2} \) | 1.3.a | |
| 5 | \( 1 + p T^{2} \) | 1.5.a | |
| 7 | \( 1 + p T^{2} \) | 1.7.a | |
| 11 | \( 1 + p T^{2} \) | 1.11.a | |
| 13 | \( 1 + p T^{2} \) | 1.13.a | |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b | |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad | |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af | |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah | |
| 31 | \( 1 + p T^{2} \) | 1.31.a | |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j | |
| 41 | \( 1 + p T^{2} \) | 1.41.a | |
| 43 | \( 1 + p T^{2} \) | 1.43.a | |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al | |
| 53 | \( 1 + p T^{2} \) | 1.53.a | |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an | |
| 61 | \( 1 + p T^{2} \) | 1.61.a | |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e | |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p | |
| 79 | \( 1 + p T^{2} \) | 1.79.a | |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai | |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r | |
| 97 | \( 1 + p T^{2} \) | 1.97.a | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.153282400834243492783638930502, −7.35667336123203915315809602260, −6.46628220799494230731841345277, −5.54555680750276191537607014291, −5.13945296261479368982339227486, −4.21486970550664934031784427234, −3.39400723683219521777097463422, −2.59992188384370109575862271106, −1.18131415567267238209540970932, 0, 1.18131415567267238209540970932, 2.59992188384370109575862271106, 3.39400723683219521777097463422, 4.21486970550664934031784427234, 5.13945296261479368982339227486, 5.54555680750276191537607014291, 6.46628220799494230731841345277, 7.35667336123203915315809602260, 8.153282400834243492783638930502