Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 2·7-s − 8-s + 9-s − 5·11-s − 2·12-s − 10·13-s + 2·14-s + 16-s − 5·17-s − 18-s − 19-s + 4·21-s + 5·22-s − 2·23-s + 2·24-s − 25-s + 10·26-s + 4·27-s − 2·28-s + 2·29-s − 4·31-s − 32-s + 10·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.577·12-s − 2.77·13-s + 0.534·14-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.229·19-s + 0.872·21-s + 1.06·22-s − 0.417·23-s + 0.408·24-s − 1/5·25-s + 1.96·26-s + 0.769·27-s − 0.377·28-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 1.74·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(129600\) = \(2^{6} \cdot 3^{4} \cdot 5^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.26340\) |
| Root analytic conductor: | \(1.69546\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 129600,\ (\ :1/2, 1/2),\ 1)\) |
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 | $C_1$ | \( 1 + T \) | |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) | ||
| 5 | $C_2$ | \( 1 + T^{2} \) | ||
| good | 7 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_a |
| 11 | $D_{4}$ | \( 1 + 5 T + 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.11.f_u | |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.k_by | |
| 17 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.17.f_bk | |
| 19 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.19.b_am | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.c_ac | |
| 29 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.29.ac_w | |
| 31 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.31.e_bc | |
| 37 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.37.m_dm | |
| 41 | $D_{4}$ | \( 1 - T + 68 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.41.ab_cq | |
| 43 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.43.af_y | |
| 47 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.47.g_bc | |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.53.e_bm | |
| 59 | $D_{4}$ | \( 1 + T - 20 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.59.b_au | |
| 61 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.61.a_c | |
| 67 | $D_{4}$ | \( 1 - 13 T + 152 T^{2} - 13 p T^{3} + p^{2} T^{4} \) | 2.67.an_fw | |
| 71 | $D_{4}$ | \( 1 + 16 T + 172 T^{2} + 16 p T^{3} + p^{2} T^{4} \) | 2.71.q_gq | |
| 73 | $D_{4}$ | \( 1 + 15 T + 120 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.73.p_eq | |
| 79 | $D_{4}$ | \( 1 - 4 T + 128 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.79.ae_ey | |
| 83 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.83.e_ck | |
| 89 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) | 2.89.a_fi | |
| 97 | $D_{4}$ | \( 1 + 7 T - 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.97.h_abc | |
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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
−14.3379022248, −13.8081131890, −13.0387453213, −12.8966841270, −12.3949624492, −12.0584064800, −11.6917629227, −11.1191494686, −10.5615080006, −10.3964500096, −9.95163632721, −9.52844205764, −8.99488329635, −8.43068081057, −7.85102408067, −7.26626760683, −7.02329433911, −6.54892442789, −5.76044278716, −5.46034416408, −4.80170226730, −4.47979350096, −3.23795484443, −2.57904139584, −2.06335025974, 0, 0, 2.06335025974, 2.57904139584, 3.23795484443, 4.47979350096, 4.80170226730, 5.46034416408, 5.76044278716, 6.54892442789, 7.02329433911, 7.26626760683, 7.85102408067, 8.43068081057, 8.99488329635, 9.52844205764, 9.95163632721, 10.3964500096, 10.5615080006, 11.1191494686, 11.6917629227, 12.0584064800, 12.3949624492, 12.8966841270, 13.0387453213, 13.8081131890, 14.3379022248