Dirichlet series
| L(s) = 1 | − 2·2-s + 4-s − 3·7-s + 2·8-s − 3·9-s + 13-s + 6·14-s − 3·16-s − 3·17-s + 6·18-s − 3·19-s − 3·23-s − 5·25-s − 2·26-s − 3·28-s − 4·31-s − 2·32-s + 6·34-s − 3·36-s − 7·37-s + 6·38-s − 2·41-s + 9·43-s + 6·46-s + 2·47-s − 49-s + 10·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.13·7-s + 0.707·8-s − 9-s + 0.277·13-s + 1.60·14-s − 3/4·16-s − 0.727·17-s + 1.41·18-s − 0.688·19-s − 0.625·23-s − 25-s − 0.392·26-s − 0.566·28-s − 0.718·31-s − 0.353·32-s + 1.02·34-s − 1/2·36-s − 1.15·37-s + 0.973·38-s − 0.312·41-s + 1.37·43-s + 0.884·46-s + 0.291·47-s − 1/7·49-s + 1.41·50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8714 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8714 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(8714\) = \(2 \cdot 4357\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.555612\) |
| Root analytic conductor: | \(0.863362\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 8714,\ (\ :1/2, 1/2),\ -1)\) |
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) | |
| 4357 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 36 T + p T^{2} ) \) | ||
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | 2.3.a_d |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | 2.5.a_f | |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.d_k | |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) | 2.11.a_af | |
| 13 | $D_{4}$ | \( 1 - T - 7 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.13.ab_ah | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.19.d_u | |
| 23 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.23.d_c | |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) | 2.29.a_b | |
| 31 | $D_{4}$ | \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.31.e_cj | |
| 37 | $D_{4}$ | \( 1 + 7 T + 68 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.37.h_cq | |
| 41 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_bn | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.43.aj_ea | |
| 47 | $D_{4}$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.47.ac_an | |
| 53 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.53.g_cg | |
| 59 | $D_{4}$ | \( 1 - 3 T - 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.59.ad_ag | |
| 61 | $D_{4}$ | \( 1 - 7 T + 19 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.61.ah_t | |
| 67 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.67.i_dt | |
| 71 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.71.e_cw | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.73.am_de | |
| 79 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) | 2.79.a_az | |
| 83 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.83.ae_w | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.89.o_gh | |
| 97 | $D_{4}$ | \( 1 - 13 T + 152 T^{2} - 13 p T^{3} + p^{2} T^{4} \) | 2.97.an_fw | |
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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
−17.1267971688, −16.7105176135, −16.0970648565, −15.8178259377, −15.2521136556, −14.4700813280, −14.0162004400, −13.5658066146, −12.8267673497, −12.6050772508, −11.6847402309, −11.2445344943, −10.5701998198, −10.1818631219, −9.55804292806, −9.02865747393, −8.71910979065, −8.06483991670, −7.47422451882, −6.66058592490, −6.08153661846, −5.36064050163, −4.18820355209, −3.37829195742, −2.12181210400, 0, 2.12181210400, 3.37829195742, 4.18820355209, 5.36064050163, 6.08153661846, 6.66058592490, 7.47422451882, 8.06483991670, 8.71910979065, 9.02865747393, 9.55804292806, 10.1818631219, 10.5701998198, 11.2445344943, 11.6847402309, 12.6050772508, 12.8267673497, 13.5658066146, 14.0162004400, 14.4700813280, 15.2521136556, 15.8178259377, 16.0970648565, 16.7105176135, 17.1267971688