Dirichlet series
| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 3·5-s + 4·6-s − 2·7-s − 4·8-s + 2·9-s + 6·10-s − 4·12-s − 2·13-s + 4·14-s + 6·15-s + 8·16-s − 4·18-s − 6·20-s + 4·21-s − 23-s + 8·24-s + 25-s + 4·26-s − 6·27-s − 4·28-s − 6·29-s − 12·30-s + 12·31-s − 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.34·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s + 1.89·10-s − 1.15·12-s − 0.554·13-s + 1.06·14-s + 1.54·15-s + 2·16-s − 0.942·18-s − 1.34·20-s + 0.872·21-s − 0.208·23-s + 1.63·24-s + 1/5·25-s + 0.784·26-s − 1.15·27-s − 0.755·28-s − 1.11·29-s − 2.19·30-s + 2.15·31-s − 1.41·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1757 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1757 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1757\) = \(7 \cdot 251\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.112027\) |
| Root analytic conductor: | \(0.578537\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 1757,\ (\ :1/2, 1/2),\ -1)\) |
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 | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) | |
| 251 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 12 T + p T^{2} ) \) | ||
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.2.c_c |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.3.c_c | |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.5.d_i | |
| 11 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) | 2.11.a_m | |
| 13 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_e | |
| 17 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) | 2.17.a_y | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 + T - 27 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.23.b_abb | |
| 29 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.29.g_w | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.31.am_de | |
| 37 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_c | |
| 41 | $D_{4}$ | \( 1 + 2 T - 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_abk | |
| 43 | $D_{4}$ | \( 1 + 11 T + 90 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.43.l_dm | |
| 47 | $D_{4}$ | \( 1 - 15 T + 129 T^{2} - 15 p T^{3} + p^{2} T^{4} \) | 2.47.ap_ez | |
| 53 | $D_{4}$ | \( 1 + 7 T + 16 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.53.h_q | |
| 59 | $D_{4}$ | \( 1 + 3 T + 89 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.59.d_dl | |
| 61 | $D_{4}$ | \( 1 + 11 T + 115 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.61.l_el | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.67.i_g | |
| 71 | $D_{4}$ | \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.71.o_ec | |
| 73 | $D_{4}$ | \( 1 + 10 T + 100 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.73.k_dw | |
| 79 | $D_{4}$ | \( 1 + 9 T + 157 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.79.j_gb | |
| 83 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.83.ai_ce | |
| 89 | $C_2^2$ | \( 1 + 172 T^{2} + p^{2} T^{4} \) | 2.89.a_gq | |
| 97 | $D_{4}$ | \( 1 - 10 T + 146 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.97.ak_fq | |
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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
−19.1121176557, −18.7600241164, −18.3765351156, −17.6016222574, −17.1826502809, −16.8828455807, −16.2322989758, −15.5904143163, −15.3681698230, −14.7858664338, −13.6449234873, −12.8521942639, −12.1433866094, −11.6818592175, −11.5657745281, −10.4649827735, −10.0661625208, −9.34412844177, −8.70571399736, −7.79682901070, −7.41131542867, −6.34386841647, −5.83587969972, −4.47478091738, −3.22884403715, 0, 3.22884403715, 4.47478091738, 5.83587969972, 6.34386841647, 7.41131542867, 7.79682901070, 8.70571399736, 9.34412844177, 10.0661625208, 10.4649827735, 11.5657745281, 11.6818592175, 12.1433866094, 12.8521942639, 13.6449234873, 14.7858664338, 15.3681698230, 15.5904143163, 16.2322989758, 16.8828455807, 17.1826502809, 17.6016222574, 18.3765351156, 18.7600241164, 19.1121176557