Dirichlet series
| L(s) = 1 | − 2-s − 3-s − 2·4-s − 4·5-s + 6-s − 7-s + 3·8-s − 2·9-s + 4·10-s − 7·11-s + 2·12-s + 14-s + 4·15-s + 16-s + 17-s + 2·18-s − 4·19-s + 8·20-s + 21-s + 7·22-s − 10·23-s − 3·24-s + 7·25-s + 5·27-s + 2·28-s − 2·29-s − 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 4-s − 1.78·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 2/3·9-s + 1.26·10-s − 2.11·11-s + 0.577·12-s + 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.917·19-s + 1.78·20-s + 0.218·21-s + 1.49·22-s − 2.08·23-s − 0.612·24-s + 7/5·25-s + 0.962·27-s + 0.377·28-s − 0.371·29-s − 0.730·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(29241\) = \(3^{4} \cdot 19^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.86443\) |
| Root analytic conductor: | \(1.16852\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 29241,\ (\ :1/2, 1/2),\ 1)\) |
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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 19 | $C_2$ | \( 1 + 4 T + p T^{2} \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.2.b_d |
| 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.5.e_j | |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_g | |
| 11 | $D_{4}$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.11.h_ba | |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) | 2.13.a_f | |
| 17 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.17.ab_y | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.23.k_cp | |
| 29 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_t | |
| 31 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.31.d_c | |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.37.a_bm | |
| 41 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.41.g_br | |
| 43 | $D_{4}$ | \( 1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.43.e_bx | |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.47.ai_r | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.53.n_fg | |
| 59 | $D_{4}$ | \( 1 - 2 T - 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.59.ac_aj | |
| 61 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.61.ac_h | |
| 67 | $D_{4}$ | \( 1 - 10 T + 83 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.67.ak_df | |
| 71 | $D_{4}$ | \( 1 - 3 T + 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.71.ad_s | |
| 73 | $D_{4}$ | \( 1 - 21 T + 252 T^{2} - 21 p T^{3} + p^{2} T^{4} \) | 2.73.av_js | |
| 79 | $D_{4}$ | \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.79.g_bz | |
| 83 | $D_{4}$ | \( 1 - T - 26 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.83.ab_aba | |
| 89 | $D_{4}$ | \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.89.ad_bs | |
| 97 | $D_{4}$ | \( 1 + 12 T + 129 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.97.m_ez | |
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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
−15.8121039280, −15.4653684435, −15.1102738004, −14.2904558756, −13.9922154519, −13.3715052463, −12.8612044283, −12.4210108615, −12.1063472131, −11.3819049826, −11.0091355591, −10.5473583295, −9.99485315928, −9.58851740111, −8.75500407212, −8.24173941425, −7.98596643088, −7.77526089182, −6.82457826540, −6.08861637757, −5.25026433595, −4.94046677881, −4.03826751350, −3.60056343405, −2.48987197505, 0, 0, 2.48987197505, 3.60056343405, 4.03826751350, 4.94046677881, 5.25026433595, 6.08861637757, 6.82457826540, 7.77526089182, 7.98596643088, 8.24173941425, 8.75500407212, 9.58851740111, 9.99485315928, 10.5473583295, 11.0091355591, 11.3819049826, 12.1063472131, 12.4210108615, 12.8612044283, 13.3715052463, 13.9922154519, 14.2904558756, 15.1102738004, 15.4653684435, 15.8121039280