Dirichlet series
| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 3·7-s + 8-s + 3·11-s + 12-s + 4·13-s + 3·14-s + 3·16-s − 4·17-s + 3·21-s − 3·22-s − 24-s − 10·25-s − 4·26-s − 2·27-s + 3·28-s − 6·29-s − 2·31-s − 3·32-s − 3·33-s + 4·34-s + 3·37-s − 4·39-s − 3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 0.904·11-s + 0.288·12-s + 1.10·13-s + 0.801·14-s + 3/4·16-s − 0.970·17-s + 0.654·21-s − 0.639·22-s − 0.204·24-s − 2·25-s − 0.784·26-s − 0.384·27-s + 0.566·28-s − 1.11·29-s − 0.359·31-s − 0.530·32-s − 0.522·33-s + 0.685·34-s + 0.493·37-s − 0.640·39-s − 0.468·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 17986 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17986 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(17986\) = \(2 \cdot 17 \cdot 23^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.14680\) |
| Root analytic conductor: | \(1.03483\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 17986,\ (\ :1/2, 1/2),\ -1)\) |
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) | ||
| 23 | $C_2$ | \( 1 + p T^{2} \) | ||
| good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_b |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.5.a_k | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.7.d_e | |
| 11 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.11.ad_h | |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.13.ae_s | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 29 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.29.g_w | |
| 31 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.31.c_bk | |
| 37 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_ai | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.41.d_bc | |
| 43 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.43.m_dw | |
| 47 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.47.ad_cs | |
| 53 | $D_{4}$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.53.j_cg | |
| 59 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.59.g_dw | |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.61.a_cg | |
| 67 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.67.d_cg | |
| 71 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) | 2.71.a_de | |
| 73 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.73.ae_ci | |
| 79 | $C_2^2$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.79.p_fy | |
| 83 | $D_{4}$ | \( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.83.aj_bu | |
| 89 | $D_{4}$ | \( 1 + 3 T - 107 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.89.d_aed | |
| 97 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.97.ag_fm | |
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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
−16.2299764048, −15.5980428722, −15.4594803961, −14.7442208257, −14.1362342532, −13.5517871938, −13.2127236386, −12.8830313665, −12.1224046793, −11.6254450524, −11.2513270024, −10.6560861922, −9.84475572901, −9.73809621825, −9.05554280059, −8.75453318562, −8.03671944910, −7.38203159937, −6.61660638593, −6.01372559543, −5.80705434103, −4.68066830339, −3.81215223341, −3.41151351102, −1.74187969815, 0, 1.74187969815, 3.41151351102, 3.81215223341, 4.68066830339, 5.80705434103, 6.01372559543, 6.61660638593, 7.38203159937, 8.03671944910, 8.75453318562, 9.05554280059, 9.73809621825, 9.84475572901, 10.6560861922, 11.2513270024, 11.6254450524, 12.1224046793, 12.8830313665, 13.2127236386, 13.5517871938, 14.1362342532, 14.7442208257, 15.4594803961, 15.5980428722, 16.2299764048