Dirichlet series
| L(s) = 1 | − 2·3-s − 4·7-s − 2·9-s + 2·11-s − 8·13-s − 2·19-s + 8·21-s + 4·23-s − 6·25-s + 10·27-s + 16·29-s − 4·33-s − 8·37-s + 16·39-s + 4·41-s − 6·43-s − 8·47-s + 2·49-s − 8·53-s + 4·57-s − 14·59-s + 8·61-s + 8·63-s − 10·67-s − 8·69-s + 12·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s − 2/3·9-s + 0.603·11-s − 2.21·13-s − 0.458·19-s + 1.74·21-s + 0.834·23-s − 6/5·25-s + 1.92·27-s + 2.97·29-s − 0.696·33-s − 1.31·37-s + 2.56·39-s + 0.624·41-s − 0.914·43-s − 1.16·47-s + 2/7·49-s − 1.09·53-s + 0.529·57-s − 1.82·59-s + 1.02·61-s + 1.00·63-s − 1.22·67-s − 0.963·69-s + 1.42·71-s + 0.936·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8192 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8192 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(8192\) = \(2^{13}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.522329\) |
| Root analytic conductor: | \(0.850131\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 8192,\ (\ :1/2, 1/2),\ -1)\) |
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 | \( 1 \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.c_g |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.a_g | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.e_o | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ac_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.i_bm | |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.17.a_be | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.19.c_bm | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.23.ae_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) | 2.29.aq_eo | |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.31.a_ck | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.i_cc | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.ae_w | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.43.g_di | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.47.i_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.53.i_w | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.59.o_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.61.ai_dy | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.67.k_fe | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.71.am_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.73.ai_ck | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.79.i_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.83.ag_gk | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.89.ai_gc | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.97.aq_gc | |
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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
−17.3115657036, −16.7248290707, −16.1152657336, −15.8880434926, −15.1228481825, −14.5925007839, −14.0930481386, −13.6430062392, −12.7248727791, −12.4433835066, −11.9637722411, −11.6664921937, −10.9094499028, −10.2342122988, −9.88634870690, −9.31735655616, −8.63593920517, −7.86574055999, −6.97464122004, −6.33050260025, −6.27282080647, −5.01452297596, −4.84373431554, −3.36730269662, −2.63895104553, 0, 2.63895104553, 3.36730269662, 4.84373431554, 5.01452297596, 6.27282080647, 6.33050260025, 6.97464122004, 7.86574055999, 8.63593920517, 9.31735655616, 9.88634870690, 10.2342122988, 10.9094499028, 11.6664921937, 11.9637722411, 12.4433835066, 12.7248727791, 13.6430062392, 14.0930481386, 14.5925007839, 15.1228481825, 15.8880434926, 16.1152657336, 16.7248290707, 17.3115657036