Dirichlet series
| L(s) = 1 | − 2-s − 4-s − 5-s − 2·7-s + 3·8-s − 3·9-s + 10-s − 6·11-s + 3·13-s + 2·14-s − 16-s − 3·17-s + 3·18-s − 4·19-s + 20-s + 6·22-s + 4·23-s + 25-s − 3·26-s + 2·28-s − 7·29-s − 10·31-s − 5·32-s + 3·34-s + 2·35-s + 3·36-s − 3·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s + 1.06·8-s − 9-s + 0.316·10-s − 1.80·11-s + 0.832·13-s + 0.534·14-s − 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.917·19-s + 0.223·20-s + 1.27·22-s + 0.834·23-s + 1/5·25-s − 0.588·26-s + 0.377·28-s − 1.29·29-s − 1.79·31-s − 0.883·32-s + 0.514·34-s + 0.338·35-s + 1/2·36-s − 0.493·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(416000\) = \(2^{8} \cdot 5^{3} \cdot 13\) |
| Sign: | $1$ |
| Analytic conductor: | \(26.5245\) |
| Root analytic conductor: | \(2.26940\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 416000,\ (\ :1/2, 1/2),\ 1)\) |
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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 5 | $C_1$ | \( 1 + T \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) | ||
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | 2.3.a_d |
| 7 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_d | |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.11.g_z | |
| 17 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.17.d_k | |
| 19 | $D_{4}$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.19.e_d | |
| 23 | $D_{4}$ | \( 1 - 4 T + 44 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.23.ae_bs | |
| 29 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.29.h_bq | |
| 31 | $D_{4}$ | \( 1 + 10 T + 76 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.31.k_cy | |
| 37 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) | 2.37.d_a | |
| 41 | $D_{4}$ | \( 1 + 9 T + 64 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.41.j_cm | |
| 43 | $D_{4}$ | \( 1 + 6 T + 87 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.43.g_dj | |
| 47 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) | 2.47.a_bp | |
| 53 | $D_{4}$ | \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.53.ag_ed | |
| 59 | $D_{4}$ | \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.59.ag_l | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.61.c_du | |
| 67 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.67.ae_q | |
| 71 | $D_{4}$ | \( 1 - 2 T - 76 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.71.ac_acy | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.73.ae_fe | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) | 2.79.o_ed | |
| 83 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.83.am_ha | |
| 89 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) | 2.89.a_aba | |
| 97 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) | 2.97.a_acd | |
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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
−13.2487305377, −12.9171838533, −12.5562127005, −11.9656792636, −11.3376254763, −10.9666808623, −10.8349494075, −10.3416237909, −9.97032531790, −9.30648627557, −9.01074526987, −8.63487317293, −8.27208410336, −7.88739172922, −7.39559049617, −6.80253173829, −6.52608257053, −5.60611385970, −5.28259402581, −5.08469496612, −3.95466288771, −3.85488623051, −3.04371999704, −2.46120434806, −1.60571699388, 0, 0, 1.60571699388, 2.46120434806, 3.04371999704, 3.85488623051, 3.95466288771, 5.08469496612, 5.28259402581, 5.60611385970, 6.52608257053, 6.80253173829, 7.39559049617, 7.88739172922, 8.27208410336, 8.63487317293, 9.01074526987, 9.30648627557, 9.97032531790, 10.3416237909, 10.8349494075, 10.9666808623, 11.3376254763, 11.9656792636, 12.5562127005, 12.9171838533, 13.2487305377