Dirichlet series
| L(s) = 1 | + 4·5-s + 8·13-s + 8·17-s + 8·25-s − 4·29-s − 6·49-s − 12·53-s + 32·65-s + 32·85-s + 8·89-s + 101-s + 103-s + 107-s + 109-s + 113-s − 18·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2.21·13-s + 1.94·17-s + 8/5·25-s − 0.742·29-s − 6/7·49-s − 1.64·53-s + 3.96·65-s + 3.47·85-s + 0.847·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.63·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(331776\) = \(2^{12} \cdot 3^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(21.1543\) |
| Root analytic conductor: | \(2.14461\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.173276169\) |
| \(L(\frac12)\) | \(\approx\) | \(3.173276169\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 3 | \( 1 \) | |||
| good | 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.5.ae_i |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) | 2.7.a_g | |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.11.a_s | |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.13.ai_bq | |
| 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.17.ai_bg | |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) | 2.19.a_be | |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.a_as | |
| 29 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.29.e_i | |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | 2.31.a_ak | |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) | 2.37.a_acg | |
| 41 | $C_2^2$ | \( 1 + p^{2} T^{4} \) | 2.41.a_a | |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) | 2.43.a_da | |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.47.a_dq | |
| 53 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.53.m_cu | |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.59.a_de | |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) | 2.61.a_aec | |
| 67 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) | 2.67.a_aco | |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.a_da | |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) | 2.73.a_abu | |
| 79 | $C_2^2$ | \( 1 + 150 T^{2} + p^{2} T^{4} \) | 2.79.a_fu | |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.83.a_fa | |
| 89 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.89.ai_bg | |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) | 2.97.a_agc | |
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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
−13.1134033273, −12.7663250821, −12.2197506970, −11.7535702866, −11.2122486187, −10.8686268323, −10.4466940745, −10.0679996556, −9.59262131863, −9.30832451479, −8.89135663765, −8.28253505043, −7.94716049506, −7.44219140220, −6.60524777327, −6.33987862173, −5.94443193555, −5.46540687554, −5.21389696946, −4.33920601107, −3.54129031628, −3.29152788264, −2.42426145010, −1.51240634341, −1.26871170597, 1.26871170597, 1.51240634341, 2.42426145010, 3.29152788264, 3.54129031628, 4.33920601107, 5.21389696946, 5.46540687554, 5.94443193555, 6.33987862173, 6.60524777327, 7.44219140220, 7.94716049506, 8.28253505043, 8.89135663765, 9.30832451479, 9.59262131863, 10.0679996556, 10.4466940745, 10.8686268323, 11.2122486187, 11.7535702866, 12.2197506970, 12.7663250821, 13.1134033273