Dirichlet series
| L(s) = 1 | + 2-s + 4-s − 3·5-s − 4·7-s + 8-s − 2·9-s − 3·10-s − 2·11-s − 3·13-s − 4·14-s + 16-s − 3·17-s − 2·18-s − 3·20-s − 2·22-s + 3·23-s + 2·25-s − 3·26-s − 3·27-s − 4·28-s + 7·29-s + 31-s + 32-s − 3·34-s + 12·35-s − 2·36-s − 7·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s − 1.51·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s − 0.603·11-s − 0.832·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.670·20-s − 0.426·22-s + 0.625·23-s + 2/5·25-s − 0.588·26-s − 0.577·27-s − 0.755·28-s + 1.29·29-s + 0.179·31-s + 0.176·32-s − 0.514·34-s + 2.02·35-s − 1/3·36-s − 1.15·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 24384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(24384\) = \(2^{6} \cdot 3 \cdot 127\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.55474\) |
| Root analytic conductor: | \(1.11664\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 24384,\ (\ :1/2, 1/2),\ -1)\) |
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) | ||
| 127 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 15 T + p T^{2} ) \) | ||
| good | 5 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.5.d_h |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.7.e_i | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.c_o | |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.13.d_r | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.19.a_c | |
| 23 | $D_{4}$ | \( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.23.ad_bj | |
| 29 | $D_{4}$ | \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.29.ah_ci | |
| 31 | $D_{4}$ | \( 1 - T - 18 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.31.ab_as | |
| 37 | $D_{4}$ | \( 1 + 7 T + 65 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.37.h_cn | |
| 41 | $D_{4}$ | \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.41.d_bk | |
| 43 | $D_{4}$ | \( 1 - 9 T + 36 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.43.aj_bk | |
| 47 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.47.g_ca | |
| 53 | $D_{4}$ | \( 1 - T - 29 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.53.ab_abd | |
| 59 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.59.ad_n | |
| 61 | $D_{4}$ | \( 1 + 9 T + 60 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.61.j_ci | |
| 67 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.67.ag_bw | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.ac_ck | |
| 73 | $D_{4}$ | \( 1 + 3 T + 125 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.73.d_ev | |
| 79 | $D_{4}$ | \( 1 - T - 13 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.79.ab_an | |
| 83 | $D_{4}$ | \( 1 + 11 T + 127 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.83.l_ex | |
| 89 | $D_{4}$ | \( 1 - T + 69 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.89.ab_cr | |
| 97 | $D_{4}$ | \( 1 + 4 T - 88 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.97.e_adk | |
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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
−15.6915821698, −15.3627546855, −14.9999859178, −14.3037862658, −13.8510951267, −13.3747930598, −12.8349470639, −12.4494846060, −12.0670199617, −11.5767194210, −11.0697143935, −10.5434975214, −9.97632289024, −9.42531601076, −8.73891490847, −8.21334200481, −7.58096542348, −7.03027166384, −6.59226516141, −5.89354890099, −5.16275846849, −4.49901998388, −3.76560047319, −3.10600494571, −2.53623303464, 0, 2.53623303464, 3.10600494571, 3.76560047319, 4.49901998388, 5.16275846849, 5.89354890099, 6.59226516141, 7.03027166384, 7.58096542348, 8.21334200481, 8.73891490847, 9.42531601076, 9.97632289024, 10.5434975214, 11.0697143935, 11.5767194210, 12.0670199617, 12.4494846060, 12.8349470639, 13.3747930598, 13.8510951267, 14.3037862658, 14.9999859178, 15.3627546855, 15.6915821698