Dirichlet series
| L(s) = 1 | − 2·5-s − 3·7-s − 9-s + 2·13-s − 4·19-s − 6·23-s − 3·25-s + 4·29-s − 5·31-s + 6·35-s − 8·37-s + 6·41-s − 8·43-s + 2·45-s − 4·47-s + 2·49-s + 2·53-s + 6·59-s − 12·61-s + 3·63-s − 4·65-s − 2·71-s + 7·79-s + 81-s + 14·83-s − 10·89-s − 6·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s − 1/3·9-s + 0.554·13-s − 0.917·19-s − 1.25·23-s − 3/5·25-s + 0.742·29-s − 0.898·31-s + 1.01·35-s − 1.31·37-s + 0.937·41-s − 1.21·43-s + 0.298·45-s − 0.583·47-s + 2/7·49-s + 0.274·53-s + 0.781·59-s − 1.53·61-s + 0.377·63-s − 0.496·65-s − 0.237·71-s + 0.787·79-s + 1/9·81-s + 1.53·83-s − 1.05·89-s − 0.628·91-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(28224\) = \(2^{6} \cdot 3^{2} \cdot 7^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.79958\) |
| Root analytic conductor: | \(1.15822\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 28224,\ (\ :1/2, 1/2),\ -1)\) |
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 | \( 1 \) | ||
| 3 | $C_2$ | \( 1 + T^{2} \) | ||
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) | ||
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.5.c_h |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) | 2.11.a_d | |
| 13 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.13.ac_n | |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | 2.17.a_ak | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.19.e_bh | |
| 23 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.23.g_o | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.29.ae_n | |
| 31 | $D_{4}$ | \( 1 + 5 T - 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.31.f_ac | |
| 37 | $D_{4}$ | \( 1 + 8 T + 31 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.37.i_bf | |
| 41 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.41.ag_cs | |
| 43 | $D_{4}$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.43.i_bv | |
| 47 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.47.e_bq | |
| 53 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.53.ac_ah | |
| 59 | $D_{4}$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.59.ag_cb | |
| 61 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.61.m_fe | |
| 67 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) | 2.67.a_dr | |
| 71 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.71.c_dy | |
| 73 | $C_2^2$ | \( 1 + 125 T^{2} + p^{2} T^{4} \) | 2.73.a_ev | |
| 79 | $D_{4}$ | \( 1 - 7 T + 78 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.79.ah_da | |
| 83 | $D_{4}$ | \( 1 - 14 T + 189 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.83.ao_hh | |
| 89 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.89.k_da | |
| 97 | $D_{4}$ | \( 1 - 8 T - 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.97.ai_al | |
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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
−15.7554950766, −15.0776450852, −14.8489167796, −14.1718495759, −13.5279172207, −13.4543128353, −12.6258169261, −12.2946067743, −11.9316522385, −11.3275545060, −10.8162447172, −10.3458009591, −9.77119123496, −9.30476018402, −8.53119511478, −8.31816709366, −7.62086327191, −7.02933562642, −6.32098537108, −6.05093239547, −5.18229918303, −4.28139829632, −3.70723788281, −3.18066973466, −2.02039137781, 0, 2.02039137781, 3.18066973466, 3.70723788281, 4.28139829632, 5.18229918303, 6.05093239547, 6.32098537108, 7.02933562642, 7.62086327191, 8.31816709366, 8.53119511478, 9.30476018402, 9.77119123496, 10.3458009591, 10.8162447172, 11.3275545060, 11.9316522385, 12.2946067743, 12.6258169261, 13.4543128353, 13.5279172207, 14.1718495759, 14.8489167796, 15.0776450852, 15.7554950766