Dirichlet series
| L(s) = 1 | + 3-s + 3·7-s + 9-s + 2·11-s − 13-s − 3·19-s + 3·21-s − 2·23-s − 2·25-s + 27-s + 6·29-s + 2·33-s + 5·37-s − 39-s − 6·41-s + 6·47-s − 5·49-s + 18·53-s − 3·57-s + 10·59-s + 61-s + 3·63-s + 9·67-s − 2·69-s + 4·71-s − 5·73-s − 2·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.688·19-s + 0.654·21-s − 0.417·23-s − 2/5·25-s + 0.192·27-s + 1.11·29-s + 0.348·33-s + 0.821·37-s − 0.160·39-s − 0.937·41-s + 0.875·47-s − 5/7·49-s + 2.47·53-s − 0.397·57-s + 1.30·59-s + 0.128·61-s + 0.377·63-s + 1.09·67-s − 0.240·69-s + 0.474·71-s − 0.585·73-s − 0.230·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 165888 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165888 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(165888\) = \(2^{11} \cdot 3^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(10.5771\) |
| Root analytic conductor: | \(1.80340\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 165888,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.411186321\) |
| \(L(\frac12)\) | \(\approx\) | \(2.411186321\) |
| \(L(\frac{3}{2})\) | 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 | $C_1$ | \( 1 - T \) | ||
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.5.a_c |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) | 2.7.ad_o | |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.11.ac_k | |
| 13 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_i | |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.17.a_c | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.d_bi | |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.23.c_c | |
| 29 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.29.ag_k | |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) | 2.31.a_be | |
| 37 | $D_{4}$ | \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.37.af_u | |
| 41 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.41.g_s | |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) | 2.43.a_cc | |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.47.ag_s | |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.53.as_gg | |
| 59 | $D_{4}$ | \( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.59.ak_ec | |
| 61 | $D_{4}$ | \( 1 - T - 76 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.61.ab_acy | |
| 67 | $D_{4}$ | \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.67.aj_cw | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.ae_bu | |
| 73 | $D_{4}$ | \( 1 + 5 T + 92 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.73.f_do | |
| 79 | $D_{4}$ | \( 1 + 3 T + 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.79.d_ck | |
| 83 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.83.ai_cs | |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) | 2.89.a_fq | |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.97.b_ga | |
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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
−13.5638533456, −13.3591590356, −12.7121568150, −12.2863266677, −11.8004834646, −11.5717507945, −11.0462963604, −10.5015415138, −10.1052280985, −9.71082147189, −9.11631492138, −8.58584376580, −8.35545937940, −7.88868604852, −7.38667699272, −6.74941998521, −6.45987133399, −5.61288286827, −5.19244772144, −4.45480143037, −4.12951489885, −3.47977361779, −2.51167541148, −2.05088101784, −1.08742327211, 1.08742327211, 2.05088101784, 2.51167541148, 3.47977361779, 4.12951489885, 4.45480143037, 5.19244772144, 5.61288286827, 6.45987133399, 6.74941998521, 7.38667699272, 7.88868604852, 8.35545937940, 8.58584376580, 9.11631492138, 9.71082147189, 10.1052280985, 10.5015415138, 11.0462963604, 11.5717507945, 11.8004834646, 12.2863266677, 12.7121568150, 13.3591590356, 13.5638533456