Dirichlet series
| L(s) = 1 | − 2·4-s + 5-s − 3·9-s − 6·11-s − 3·13-s + 4·16-s − 7·17-s − 2·20-s + 2·23-s + 3·25-s − 3·29-s − 6·31-s + 6·36-s + 37-s − 4·41-s + 6·43-s + 12·44-s − 3·45-s + 6·47-s + 2·49-s + 6·52-s − 6·53-s − 6·55-s + 16·59-s − 7·61-s − 8·64-s − 3·65-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 9-s − 1.80·11-s − 0.832·13-s + 16-s − 1.69·17-s − 0.447·20-s + 0.417·23-s + 3/5·25-s − 0.557·29-s − 1.07·31-s + 36-s + 0.164·37-s − 0.624·41-s + 0.914·43-s + 1.80·44-s − 0.447·45-s + 0.875·47-s + 2/7·49-s + 0.832·52-s − 0.824·53-s − 0.809·55-s + 2.08·59-s − 0.896·61-s − 64-s − 0.372·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(20736\) = \(2^{8} \cdot 3^{4}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.32214\) |
| Root analytic conductor: | \(1.07230\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 20736,\ (\ :1/2, 1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) | |
| 3 | $C_2$ | \( 1 + p T^{2} \) | ||
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.5.ab_ac |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.a_ac | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.g_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.13.d_w | |
| 17 | $D_{4}$ | \( 1 + 7 T + 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.17.h_bc | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.ac_ac | |
| 29 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.29.d_w | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.g_bu | |
| 37 | $D_{4}$ | \( 1 - T - 18 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.37.ab_as | |
| 41 | $D_{4}$ | \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.41.e_aba | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.43.ag_cs | |
| 47 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.47.ag_ac | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.53.g_bi | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.59.aq_gk | |
| 61 | $D_{4}$ | \( 1 + 7 T + 110 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.61.h_eg | |
| 67 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.67.ai_es | |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) | 2.71.a_aby | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.73.af_dc | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.79.c_da | |
| 83 | $D_{4}$ | \( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.83.k_ec | |
| 89 | $D_{4}$ | \( 1 + 19 T + 208 T^{2} + 19 p T^{3} + p^{2} T^{4} \) | 2.89.t_ia | |
| 97 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.97.m_eo | |
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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
−15.8770442507, −15.4040036954, −14.9089107741, −14.5600222105, −13.8473792525, −13.6874501329, −13.0426079089, −12.7476272942, −12.3866407162, −11.4429292236, −10.9518997444, −10.6769413635, −9.92122934179, −9.52848749240, −8.83273169920, −8.59712227680, −7.91354714015, −7.32315173760, −6.65238832820, −5.60799740615, −5.42814910896, −4.80091959668, −4.02415785541, −2.89684943383, −2.30913979404, 0, 2.30913979404, 2.89684943383, 4.02415785541, 4.80091959668, 5.42814910896, 5.60799740615, 6.65238832820, 7.32315173760, 7.91354714015, 8.59712227680, 8.83273169920, 9.52848749240, 9.92122934179, 10.6769413635, 10.9518997444, 11.4429292236, 12.3866407162, 12.7476272942, 13.0426079089, 13.6874501329, 13.8473792525, 14.5600222105, 14.9089107741, 15.4040036954, 15.8770442507