Dirichlet series
| L(s) = 1 | + 2·5-s + 2·9-s − 6·13-s + 2·25-s − 6·29-s + 2·37-s + 4·45-s + 2·49-s + 2·53-s − 6·61-s − 12·65-s + 4·73-s − 5·81-s − 28·89-s + 101-s + 103-s + 107-s + 109-s + 113-s − 12·117-s − 14·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2/3·9-s − 1.66·13-s + 2/5·25-s − 1.11·29-s + 0.328·37-s + 0.596·45-s + 2/7·49-s + 0.274·53-s − 0.768·61-s − 1.48·65-s + 0.468·73-s − 5/9·81-s − 2.96·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.10·117-s − 1.27·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8192 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8192 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(8192\) = \(2^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.522329\) |
| Root analytic conductor: | \(0.850131\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 8192,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.053463345\) |
| \(L(\frac12)\) | \(\approx\) | \(1.053463345\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|---|
| bad | 2 | \( 1 \) | ||
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.3.a_ac |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.ac_c | |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.a_ac | |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) | 2.11.a_o | |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.13.g_s | |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | 2.17.a_abi | |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) | 2.19.a_be | |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.23.a_be | |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.29.g_s | |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.a_ac | |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.37.ac_c | |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.41.a_as | |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) | 2.43.a_da | |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.47.a_be | |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.53.ac_c | |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) | 2.59.a_bu | |
| 61 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.61.g_s | |
| 67 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) | 2.67.a_ew | |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.71.a_ac | |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.73.ae_fu | |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.79.a_gc | |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) | 2.83.a_abi | |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) | 2.89.bc_ok | |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) | 2.97.a_ck | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.0069330445, −16.5381033884, −15.8251582292, −15.2196059936, −14.9345141563, −14.2722229092, −13.9095518160, −13.2949725176, −12.6704011920, −12.5210069653, −11.7245156452, −11.1653383253, −10.4390913372, −9.92866419805, −9.59768506301, −9.04041612201, −8.22929165956, −7.36382185165, −7.14289589908, −6.22533396944, −5.53178318754, −4.87150278747, −4.08772000384, −2.83435142025, −1.87577862067, 1.87577862067, 2.83435142025, 4.08772000384, 4.87150278747, 5.53178318754, 6.22533396944, 7.14289589908, 7.36382185165, 8.22929165956, 9.04041612201, 9.59768506301, 9.92866419805, 10.4390913372, 11.1653383253, 11.7245156452, 12.5210069653, 12.6704011920, 13.2949725176, 13.9095518160, 14.2722229092, 14.9345141563, 15.2196059936, 15.8251582292, 16.5381033884, 17.0069330445