Dirichlet series
| L(s) = 1 | + 2-s − 2·3-s + 3·5-s − 2·6-s − 4·7-s − 8-s + 2·9-s + 3·10-s + 5·11-s − 8·13-s − 4·14-s − 6·15-s − 16-s + 2·18-s − 4·19-s + 8·21-s + 5·22-s − 3·23-s + 2·24-s − 2·25-s − 8·26-s − 6·27-s + 7·29-s − 6·30-s − 2·31-s − 10·33-s − 12·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1.34·5-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 2/3·9-s + 0.948·10-s + 1.50·11-s − 2.21·13-s − 1.06·14-s − 1.54·15-s − 1/4·16-s + 0.471·18-s − 0.917·19-s + 1.74·21-s + 1.06·22-s − 0.625·23-s + 0.408·24-s − 2/5·25-s − 1.56·26-s − 1.15·27-s + 1.29·29-s − 1.09·30-s − 0.359·31-s − 1.74·33-s − 2.02·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 58492 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58492 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(58492\) = \(2^{2} \cdot 7 \cdot 2089\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.72950\) |
| Root analytic conductor: | \(1.38967\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 58492,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 - T + T^{2} \) | |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) | ||
| 2089 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 39 T + p T^{2} ) \) | ||
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.3.c_c |
| 5 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.5.ad_l | |
| 11 | $D_{4}$ | \( 1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.11.af_t | |
| 13 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.13.i_bg | |
| 17 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) | 2.17.a_x | |
| 19 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.19.e_p | |
| 23 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.23.d_q | |
| 29 | $D_{4}$ | \( 1 - 7 T + 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.29.ah_w | |
| 31 | $D_{4}$ | \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.31.c_br | |
| 37 | $D_{4}$ | \( 1 + 7 T + 59 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.37.h_ch | |
| 41 | $D_{4}$ | \( 1 - 10 T + 85 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.41.ak_dh | |
| 43 | $D_{4}$ | \( 1 + 3 T + 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.43.d_bm | |
| 47 | $D_{4}$ | \( 1 + 10 T + 107 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.47.k_ed | |
| 53 | $D_{4}$ | \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.53.ad_cu | |
| 59 | $D_{4}$ | \( 1 - 4 T + 85 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.59.ae_dh | |
| 61 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.61.g_cg | |
| 67 | $D_{4}$ | \( 1 + 5 T + 129 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.67.f_ez | |
| 71 | $D_{4}$ | \( 1 - 15 T + 114 T^{2} - 15 p T^{3} + p^{2} T^{4} \) | 2.71.ap_ek | |
| 73 | $D_{4}$ | \( 1 + T - 95 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.73.b_adr | |
| 79 | $D_{4}$ | \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.79.k_ff | |
| 83 | $D_{4}$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.83.ab_ak | |
| 89 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) | 2.89.a_abl | |
| 97 | $D_{4}$ | \( 1 - 11 T + 57 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.97.al_cf | |
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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
−14.5298365216, −14.4149365414, −13.8694444256, −13.4207720782, −12.8877880759, −12.6240377534, −12.1338212904, −11.8479858489, −11.4194026025, −10.5236035015, −10.1504927632, −9.75888703373, −9.42136980694, −9.12165913226, −8.15974908993, −7.29532939861, −6.77256510940, −6.31868425472, −6.09547707277, −5.49353323477, −4.94008403564, −4.25780649847, −3.63285153115, −2.63484404000, −1.87603141293, 0, 1.87603141293, 2.63484404000, 3.63285153115, 4.25780649847, 4.94008403564, 5.49353323477, 6.09547707277, 6.31868425472, 6.77256510940, 7.29532939861, 8.15974908993, 9.12165913226, 9.42136980694, 9.75888703373, 10.1504927632, 10.5236035015, 11.4194026025, 11.8479858489, 12.1338212904, 12.6240377534, 12.8877880759, 13.4207720782, 13.8694444256, 14.4149365414, 14.5298365216