Dirichlet series
| L(s) = 1 | − 2·4-s + 2·5-s − 4·7-s − 2·9-s + 8·11-s − 8·13-s + 4·16-s − 6·17-s − 4·20-s + 8·23-s − 3·25-s + 8·28-s + 4·29-s − 8·35-s + 4·36-s + 12·37-s + 4·41-s − 16·44-s − 4·45-s − 4·47-s − 49-s + 16·52-s + 8·53-s + 16·55-s − 14·61-s + 8·63-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s − 1.51·7-s − 2/3·9-s + 2.41·11-s − 2.21·13-s + 16-s − 1.45·17-s − 0.894·20-s + 1.66·23-s − 3/5·25-s + 1.51·28-s + 0.742·29-s − 1.35·35-s + 2/3·36-s + 1.97·37-s + 0.624·41-s − 2.41·44-s − 0.596·45-s − 0.583·47-s − 1/7·49-s + 2.21·52-s + 1.09·53-s + 2.15·55-s − 1.79·61-s + 1.00·63-s − 64-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
| \(L(1)\) | \(\approx\) | \(0.5033013528\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5033013528\) |
| \(L(\frac{3}{2})\) | 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 | $C_2$ | \( 1 + p T^{2} \) | |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.a_c |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.5.ac_h | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.7.e_r | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.11.ai_bl | |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.13.i_bq | |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) | 2.17.g_br | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.23.ai_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.ae_bu | |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.31.a_bu | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.37.am_dq | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.ae_w | |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.43.a_dh | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.47.e_dt | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.53.ai_cg | |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.59.a_de | |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.61.o_ff | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.q_ha | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.71.ai_fy | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.73.ac_df | |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | 2.79.aq_io | |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.83.a_w | |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) | 2.89.ay_mk | |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.97.a_fa | |
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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
−19.5247104264, −19.1335612046, −18.1213987035, −17.4264315599, −17.4072402996, −16.7022471733, −16.5235345977, −15.2422541929, −14.6393155115, −14.5011418501, −13.5235052231, −13.3514918073, −12.5281782077, −12.0629600164, −11.3194473714, −10.2432044592, −9.55332854417, −9.18016854907, −9.08456140983, −7.62429229152, −6.44323514092, −6.39084306103, −5.03912355415, −4.15537276956, −2.82511261071, 2.82511261071, 4.15537276956, 5.03912355415, 6.39084306103, 6.44323514092, 7.62429229152, 9.08456140983, 9.18016854907, 9.55332854417, 10.2432044592, 11.3194473714, 12.0629600164, 12.5281782077, 13.3514918073, 13.5235052231, 14.5011418501, 14.6393155115, 15.2422541929, 16.5235345977, 16.7022471733, 17.4072402996, 17.4264315599, 18.1213987035, 19.1335612046, 19.5247104264