Dirichlet series
| L(s) = 1 | + 4-s − 4·5-s + 2·7-s − 5·9-s − 4·11-s + 4·13-s + 16-s + 6·17-s − 4·20-s + 2·23-s + 6·25-s + 2·28-s + 4·29-s − 12·31-s − 8·35-s − 5·36-s − 8·41-s + 12·43-s − 4·44-s + 20·45-s + 8·47-s − 7·49-s + 4·52-s − 4·53-s + 16·55-s + 24·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.78·5-s + 0.755·7-s − 5/3·9-s − 1.20·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.894·20-s + 0.417·23-s + 6/5·25-s + 0.377·28-s + 0.742·29-s − 2.15·31-s − 1.35·35-s − 5/6·36-s − 1.24·41-s + 1.82·43-s − 0.603·44-s + 2.98·45-s + 1.16·47-s − 49-s + 0.554·52-s − 0.549·53-s + 2.15·55-s + 3.12·59-s − 1.02·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1444\) = \(2^{2} \cdot 19^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0920706\) |
| Root analytic conductor: | \(0.550846\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 1444,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5162973993\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5162973993\) |
| \(L(\frac{3}{2})\) | 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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.3.a_f |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.7.ac_l | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.e_k | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.13.ae_v | |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) | 2.17.ag_br | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.23.ac_br | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.29.ae_n | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.m_dq | |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.37.a_cs | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.41.i_de | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.43.am_eo | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.ai_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.53.e_ef | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) | 2.59.ay_jt | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.i_dy | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.67.ai_ft | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.71.e_fa | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.73.ac_df | |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) | 2.79.u_jy | |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.83.m_hu | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.89.m_gw | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.97.m_ig | |
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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
−19.5389705376, −18.7224768825, −18.6052503005, −17.6870194097, −17.2033309380, −16.2774200481, −16.2230215667, −15.5135452227, −15.0017429576, −14.3734863297, −14.0221692267, −12.9067116798, −12.3882574259, −11.6151508324, −11.2480762920, −10.9885026417, −10.0989747998, −8.68441372039, −8.40848690901, −7.73591843743, −7.25038064773, −5.81027585653, −5.28069074518, −3.86057015395, −2.99432139105, 2.99432139105, 3.86057015395, 5.28069074518, 5.81027585653, 7.25038064773, 7.73591843743, 8.40848690901, 8.68441372039, 10.0989747998, 10.9885026417, 11.2480762920, 11.6151508324, 12.3882574259, 12.9067116798, 14.0221692267, 14.3734863297, 15.0017429576, 15.5135452227, 16.2230215667, 16.2774200481, 17.2033309380, 17.6870194097, 18.6052503005, 18.7224768825, 19.5389705376