Dirichlet series
| L(s) = 1 | − 2·2-s − 3·3-s + 6·6-s − 7-s + 4·8-s + 6·9-s − 5·11-s − 13-s + 2·14-s − 4·16-s + 4·17-s − 12·18-s + 19-s + 3·21-s + 10·22-s − 12·24-s + 25-s + 2·26-s − 9·27-s + 5·29-s − 8·31-s + 15·33-s − 8·34-s − 3·37-s − 2·38-s + 3·39-s + 5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 2.44·6-s − 0.377·7-s + 1.41·8-s + 2·9-s − 1.50·11-s − 0.277·13-s + 0.534·14-s − 16-s + 0.970·17-s − 2.82·18-s + 0.229·19-s + 0.654·21-s + 2.13·22-s − 2.44·24-s + 1/5·25-s + 0.392·26-s − 1.73·27-s + 0.928·29-s − 1.43·31-s + 2.61·33-s − 1.37·34-s − 0.493·37-s − 0.324·38-s + 0.480·39-s + 0.780·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(731025\) = \(3^{4} \cdot 5^{2} \cdot 19^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(46.6107\) |
| Root analytic conductor: | \(2.61289\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 731025,\ (\ :1/2, 1/2),\ 1)\) |
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 | 3 | $C_2$ | \( 1 + p T + p T^{2} \) | |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| 19 | $C_2$ | \( 1 - T + p T^{2} \) | ||
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.2.c_e |
| 7 | $C_4$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_f | |
| 11 | $D_{4}$ | \( 1 + 5 T + 13 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.11.f_n | |
| 13 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_p | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.17.ae_bi | |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.a_k | |
| 29 | $D_{4}$ | \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.29.af_bl | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.i_ck | |
| 37 | $D_{4}$ | \( 1 + 3 T + 61 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.37.d_cj | |
| 41 | $D_{4}$ | \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.41.af_n | |
| 43 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.43.g_bc | |
| 47 | $D_{4}$ | \( 1 + 14 T + 103 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.47.o_dz | |
| 53 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_ac | |
| 59 | $D_{4}$ | \( 1 + 14 T + 112 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.59.o_ei | |
| 61 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.61.m_de | |
| 67 | $D_{4}$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.67.i_bp | |
| 71 | $D_{4}$ | \( 1 + 7 T + 25 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.71.h_z | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) | 2.73.m_cj | |
| 79 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.79.i_es | |
| 83 | $D_{4}$ | \( 1 + 3 T - 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.83.d_abv | |
| 89 | $D_{4}$ | \( 1 - 6 T + 136 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.89.ag_fg | |
| 97 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.97.am_hu | |
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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
−12.6841821374, −12.2745689650, −11.9068217858, −11.4483097383, −10.9649589306, −10.6217574602, −10.3029039188, −9.97341081999, −9.78762338022, −9.16992047580, −8.85403739370, −8.27109871522, −7.78964353793, −7.50282307974, −7.20668928957, −6.43119713842, −6.09791352074, −5.57072993674, −5.13464222888, −4.64570704253, −4.49582165875, −3.35884277758, −2.99441890387, −1.75107483705, −1.18099127371, 0, 0, 1.18099127371, 1.75107483705, 2.99441890387, 3.35884277758, 4.49582165875, 4.64570704253, 5.13464222888, 5.57072993674, 6.09791352074, 6.43119713842, 7.20668928957, 7.50282307974, 7.78964353793, 8.27109871522, 8.85403739370, 9.16992047580, 9.78762338022, 9.97341081999, 10.3029039188, 10.6217574602, 10.9649589306, 11.4483097383, 11.9068217858, 12.2745689650, 12.6841821374