Dirichlet series
| L(s) = 1 | − 2-s − 4·5-s − 3·7-s − 8-s − 9-s + 4·10-s − 11-s − 4·13-s + 3·14-s − 16-s − 2·17-s + 18-s − 2·19-s + 22-s + 4·25-s + 4·26-s − 7·29-s − 4·31-s + 6·32-s + 2·34-s + 12·35-s − 37-s + 2·38-s + 4·40-s + 7·41-s − 14·43-s + 4·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·5-s − 1.13·7-s − 0.353·8-s − 1/3·9-s + 1.26·10-s − 0.301·11-s − 1.10·13-s + 0.801·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.458·19-s + 0.213·22-s + 4/5·25-s + 0.784·26-s − 1.29·29-s − 0.718·31-s + 1.06·32-s + 0.342·34-s + 2.02·35-s − 0.164·37-s + 0.324·38-s + 0.632·40-s + 1.09·41-s − 2.13·43-s + 0.596·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 87093 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87093 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(87093\) = \(3^{2} \cdot 9677\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.55312\) |
| Root analytic conductor: | \(1.53509\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 87093,\ (\ :1/2, 1/2),\ 1)\) |
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 | 3 | $C_2$ | \( 1 + T^{2} \) | |
| 9677 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 128 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) | 2.2.b_b |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.5.e_m | |
| 7 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.7.d_n | |
| 11 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.11.b_j | |
| 13 | $D_{4}$ | \( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.13.e_t | |
| 17 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.17.c_y | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.19.c_x | |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | 2.23.a_ak | |
| 29 | $D_{4}$ | \( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.29.h_bn | |
| 31 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.31.e_s | |
| 37 | $D_{4}$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.37.b_abe | |
| 41 | $D_{4}$ | \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.41.ah_dj | |
| 43 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.43.o_eo | |
| 47 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.47.ac_abi | |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) | 2.53.a_cg | |
| 59 | $D_{4}$ | \( 1 + 10 T + 120 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.59.k_eq | |
| 61 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.61.f_bk | |
| 67 | $D_{4}$ | \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.67.ah_ci | |
| 71 | $D_{4}$ | \( 1 - 11 T + 133 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.71.al_fd | |
| 73 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.73.d_d | |
| 79 | $D_{4}$ | \( 1 - 5 T - 25 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.79.af_az | |
| 83 | $D_{4}$ | \( 1 - 6 T - 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.83.ag_ak | |
| 89 | $D_{4}$ | \( 1 + 11 T + 80 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.89.l_dc | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.97.m_gz | |
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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
−14.8816321747, −14.3327038389, −13.6019727336, −13.2904693289, −12.6825750535, −12.3600723449, −11.9956599091, −11.5043692572, −10.9643148214, −10.8520052796, −9.83035488895, −9.63834221136, −9.28199070972, −8.60453756314, −8.20472965032, −7.78153649061, −7.25280896452, −6.80174383092, −6.27330435964, −5.52685072952, −4.79561167900, −4.19310175144, −3.56425247428, −3.03859381502, −2.16127736836, 0, 0, 2.16127736836, 3.03859381502, 3.56425247428, 4.19310175144, 4.79561167900, 5.52685072952, 6.27330435964, 6.80174383092, 7.25280896452, 7.78153649061, 8.20472965032, 8.60453756314, 9.28199070972, 9.63834221136, 9.83035488895, 10.8520052796, 10.9643148214, 11.5043692572, 11.9956599091, 12.3600723449, 12.6825750535, 13.2904693289, 13.6019727336, 14.3327038389, 14.8816321747