Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 4·7-s − 8-s − 2·9-s − 4·11-s − 2·12-s − 13-s + 4·14-s + 16-s + 4·17-s + 2·18-s + 8·21-s + 4·22-s − 2·23-s + 2·24-s − 25-s + 26-s + 10·27-s − 4·28-s + 2·29-s − 32-s + 8·33-s − 4·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s − 0.353·8-s − 2/3·9-s − 1.20·11-s − 0.577·12-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.471·18-s + 1.74·21-s + 0.852·22-s − 0.417·23-s + 0.408·24-s − 1/5·25-s + 0.196·26-s + 1.92·27-s − 0.755·28-s + 0.371·29-s − 0.176·32-s + 1.39·33-s − 0.685·34-s − 1/3·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 5032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(5032\) = \(2^{3} \cdot 17 \cdot 37\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.320844\) |
| Root analytic conductor: | \(0.752616\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 5032,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 + T \) | |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) | ||
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.c_g |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.5.a_b | |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.7.e_i | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.e_k | |
| 13 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_i | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.23.c_af | |
| 29 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.29.ac_h | |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) | 2.31.a_t | |
| 41 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.41.d_ac | |
| 43 | $D_{4}$ | \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.43.k_cq | |
| 47 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) | 2.47.a_cp | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.53.ae_k | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.59.ai_bi | |
| 61 | $D_{4}$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.61.aj_bc | |
| 67 | $D_{4}$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.67.g_n | |
| 71 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) | 2.71.a_ace | |
| 73 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.73.f_y | |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) | 2.79.a_de | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) | 2.83.ae_b | |
| 89 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.89.d_e | |
| 97 | $D_{4}$ | \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.97.ai_cw | |
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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
−17.6817050552, −17.2528491445, −16.6794036778, −16.3965973303, −15.9872668627, −15.4850978680, −14.8040394701, −14.1536903226, −13.4462003659, −12.9590931450, −12.2401095155, −11.9375834951, −11.4167159139, −10.6061046106, −10.1642891477, −9.92889959328, −8.95597496167, −8.39604731962, −7.67253625742, −6.80400320307, −6.34432836960, −5.50544011548, −5.22300502864, −3.51536963245, −2.69540481655, 0, 2.69540481655, 3.51536963245, 5.22300502864, 5.50544011548, 6.34432836960, 6.80400320307, 7.67253625742, 8.39604731962, 8.95597496167, 9.92889959328, 10.1642891477, 10.6061046106, 11.4167159139, 11.9375834951, 12.2401095155, 12.9590931450, 13.4462003659, 14.1536903226, 14.8040394701, 15.4850978680, 15.9872668627, 16.3965973303, 16.6794036778, 17.2528491445, 17.6817050552