Dirichlet series
| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 3·7-s − 8-s − 2·9-s − 10-s + 4·11-s − 12-s + 13-s + 3·14-s − 15-s + 3·16-s − 3·17-s − 2·18-s + 19-s + 20-s + 3·21-s + 4·22-s − 3·23-s − 24-s + 3·25-s + 26-s − 2·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 3/4·16-s − 0.727·17-s − 0.471·18-s + 0.229·19-s + 0.223·20-s + 0.654·21-s + 0.852·22-s − 0.625·23-s − 0.204·24-s + 3/5·25-s + 0.196·26-s − 0.384·27-s − 0.566·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 24398 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24398 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(24398\) = \(2 \cdot 11 \cdot 1109\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.55563\) |
| Root analytic conductor: | \(1.11680\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 24398,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.831382054\) |
| \(L(\frac12)\) | \(\approx\) | \(1.831382054\) |
| \(L(\frac{3}{2})\) | 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$ | \( ( 1 - T )( 1 + p T^{2} ) \) | |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) | ||
| 1109 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 24 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) | 2.3.ab_d |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.b_ac | |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.7.ad_k | |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.13.ab_am | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.19.ab_j | |
| 23 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.23.d_bo | |
| 29 | $D_{4}$ | \( 1 + T - 35 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.29.b_abj | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.af_ck | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.i_cc | |
| 41 | $D_{4}$ | \( 1 - 11 T + 64 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.41.al_cm | |
| 43 | $D_{4}$ | \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.43.c_ci | |
| 47 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.47.ag_cg | |
| 53 | $D_{4}$ | \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.53.h_bi | |
| 59 | $D_{4}$ | \( 1 + 9 T + 85 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.59.j_dh | |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) | 2.61.a_aby | |
| 67 | $D_{4}$ | \( 1 + 9 T + 118 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.67.j_eo | |
| 71 | $D_{4}$ | \( 1 - 4 T + 4 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.71.ae_e | |
| 73 | $D_{4}$ | \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.73.d_bl | |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) | 2.79.a_ao | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.83.an_hu | |
| 89 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.89.i_ca | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.97.ac_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
−15.3475728396, −14.7865228846, −14.4326882239, −14.2341435135, −13.6761307278, −13.5145016378, −12.6360305247, −12.2041650103, −11.8589018883, −11.2103039148, −10.8988080756, −10.2514411319, −9.34402597073, −9.08319561790, −8.50721538906, −8.06644537853, −7.62018812321, −6.75086030705, −6.11256853523, −5.42099707267, −4.68378367018, −4.25913916840, −3.62846089870, −2.78500413729, −1.49740431126, 1.49740431126, 2.78500413729, 3.62846089870, 4.25913916840, 4.68378367018, 5.42099707267, 6.11256853523, 6.75086030705, 7.62018812321, 8.06644537853, 8.50721538906, 9.08319561790, 9.34402597073, 10.2514411319, 10.8988080756, 11.2103039148, 11.8589018883, 12.2041650103, 12.6360305247, 13.5145016378, 13.6761307278, 14.2341435135, 14.4326882239, 14.7865228846, 15.3475728396