Dirichlet series
| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 2·5-s + 4·6-s + 2·7-s + 9-s + 4·10-s − 2·11-s − 4·12-s − 2·13-s − 4·14-s + 4·15-s − 4·16-s + 2·17-s − 2·18-s + 2·19-s − 4·20-s − 4·21-s + 4·22-s + 8·23-s − 6·25-s + 4·26-s − 2·27-s + 4·28-s + 4·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s + 0.755·7-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 1.15·12-s − 0.554·13-s − 1.06·14-s + 1.03·15-s − 16-s + 0.485·17-s − 0.471·18-s + 0.458·19-s − 0.894·20-s − 0.872·21-s + 0.852·22-s + 1.66·23-s − 6/5·25-s + 0.784·26-s − 0.384·27-s + 0.755·28-s + 0.742·29-s − 1.46·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(21904\) = \(2^{4} \cdot 37^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.39661\) |
| Root analytic conductor: | \(1.08709\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 21904,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + p T + p T^{2} \) | |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.c_d |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.c_k | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.7.ac_l | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.11.c_h | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.13.c_ba | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.17.ac_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.19.ac_bm | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.23.ai_cg | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.ae_bu | |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.31.a_bu | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.41.c_t | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.c_da | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.47.k_dz | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) | 2.53.ak_el | |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.59.a_cc | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.61.m_fy | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.e_bm | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) | 2.71.ao_hf | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.73.o_gd | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.79.ao_hq | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.83.o_fv | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.89.ac_go | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.97.i_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
−16.1920174169, −15.6038578732, −15.0898133484, −14.7795750605, −13.9857216787, −13.4242388779, −12.9583864139, −12.1179230334, −11.7573247228, −11.5526346186, −10.8240520925, −10.7751381625, −9.93309835361, −9.59657912559, −8.90499380067, −8.08033761001, −8.01433080787, −7.36102788146, −6.87039121695, −6.07828897597, −5.10085747285, −5.00317001401, −3.97768080475, −2.81288670918, −1.43453783212, 0, 1.43453783212, 2.81288670918, 3.97768080475, 5.00317001401, 5.10085747285, 6.07828897597, 6.87039121695, 7.36102788146, 8.01433080787, 8.08033761001, 8.90499380067, 9.59657912559, 9.93309835361, 10.7751381625, 10.8240520925, 11.5526346186, 11.7573247228, 12.1179230334, 12.9583864139, 13.4242388779, 13.9857216787, 14.7795750605, 15.0898133484, 15.6038578732, 16.1920174169