Dirichlet series
| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s − 2·7-s + 9-s + 8·10-s − 6·11-s − 4·12-s − 8·13-s + 4·14-s + 8·15-s − 4·16-s − 4·17-s − 2·18-s + 8·19-s − 8·20-s + 4·21-s + 12·22-s − 4·23-s + 2·25-s + 16·26-s − 2·27-s − 4·28-s + 8·29-s − 16·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s − 0.755·7-s + 1/3·9-s + 2.52·10-s − 1.80·11-s − 1.15·12-s − 2.21·13-s + 1.06·14-s + 2.06·15-s − 16-s − 0.970·17-s − 0.471·18-s + 1.83·19-s − 1.78·20-s + 0.872·21-s + 2.55·22-s − 0.834·23-s + 2/5·25-s + 3.13·26-s − 0.384·27-s − 0.755·28-s + 1.48·29-s − 2.92·30-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: | \(2\) |
| 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 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.c_d |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.5.e_o | |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) | 2.7.c_p | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.11.g_bb | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.i_bm | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.17.e_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.19.ai_bm | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.e_bi | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.29.ai_cs | |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.43.a_de | |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) | 2.47.s_gt | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.53.c_dz | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) | 2.59.au_ig | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.61.e_dm | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.67.ai_fe | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.71.ac_db | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.73.ag_fj | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.79.ae_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.83.s_id | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.89.i_fa | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.97.e_gg | |
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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, −15.8576685927, −15.6038578732, −15.1016494290, −14.3087253085, −13.7052370125, −12.9583864139, −12.6720749304, −11.8110660637, −11.7573247228, −11.3825198161, −10.7751381625, −9.96956258205, −9.93309835361, −9.49466349058, −8.28916011697, −8.01433080787, −7.75456110221, −6.87039121695, −6.86682095955, −5.55163436816, −5.00317001401, −4.49161097993, −3.32391285922, −2.42130817561, 0, 0, 2.42130817561, 3.32391285922, 4.49161097993, 5.00317001401, 5.55163436816, 6.86682095955, 6.87039121695, 7.75456110221, 8.01433080787, 8.28916011697, 9.49466349058, 9.93309835361, 9.96956258205, 10.7751381625, 11.3825198161, 11.7573247228, 11.8110660637, 12.6720749304, 12.9583864139, 13.7052370125, 14.3087253085, 15.1016494290, 15.6038578732, 15.8576685927, 16.1920174169