Dirichlet series
| L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s + 7-s − 3·8-s − 2·9-s − 10-s − 9·11-s − 2·12-s + 4·13-s + 14-s − 2·15-s − 16-s − 3·17-s − 2·18-s − 2·19-s + 20-s + 2·21-s − 9·22-s − 4·23-s − 6·24-s − 3·25-s + 4·26-s − 10·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s − 1.06·8-s − 2/3·9-s − 0.316·10-s − 2.71·11-s − 0.577·12-s + 1.10·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.458·19-s + 0.223·20-s + 0.436·21-s − 1.91·22-s − 0.834·23-s − 1.22·24-s − 3/5·25-s + 0.784·26-s − 1.92·27-s − 0.188·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 63296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(63296\) = \(2^{6} \cdot 23 \cdot 43\) |
| Sign: | $-1$ |
| Analytic conductor: | \(4.03580\) |
| Root analytic conductor: | \(1.41736\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 63296,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 - T + p T^{2} \) | |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) | ||
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.3.ac_g |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.5.b_e | |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.7.ab_d | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.j_bo | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.13.ae_ba | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.19.c_c | |
| 29 | $D_{4}$ | \( 1 + 5 T + 10 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.29.f_k | |
| 31 | $C_4$ | \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.31.ae_ag | |
| 37 | $D_{4}$ | \( 1 - T - 31 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.37.ab_abf | |
| 41 | $D_{4}$ | \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.41.aj_br | |
| 47 | $D_{4}$ | \( 1 - 3 T + 73 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.47.ad_cv | |
| 53 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.53.ag_ca | |
| 59 | $D_{4}$ | \( 1 + 7 T + 115 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.59.h_el | |
| 61 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.61.g_bi | |
| 67 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) | 2.67.a_adc | |
| 71 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.71.c_ac | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.73.aj_bi | |
| 79 | $D_{4}$ | \( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.79.e_dc | |
| 83 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.83.e_ac | |
| 89 | $D_{4}$ | \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.89.p_ho | |
| 97 | $D_{4}$ | \( 1 + T + 158 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.97.b_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
−14.6754076184, −14.0809753901, −13.7814035391, −13.5929955372, −13.1029752317, −12.6409666434, −12.3275450103, −11.4759307572, −11.0756622853, −10.8361805174, −10.1200116900, −9.54990403384, −8.92849187023, −8.58627789228, −8.16898704160, −7.79082749223, −7.45706361994, −6.20942943732, −5.75770837395, −5.46061700532, −4.57707030987, −4.08683715775, −3.41940832811, −2.61724731092, −2.40201091097, 0, 2.40201091097, 2.61724731092, 3.41940832811, 4.08683715775, 4.57707030987, 5.46061700532, 5.75770837395, 6.20942943732, 7.45706361994, 7.79082749223, 8.16898704160, 8.58627789228, 8.92849187023, 9.54990403384, 10.1200116900, 10.8361805174, 11.0756622853, 11.4759307572, 12.3275450103, 12.6409666434, 13.1029752317, 13.5929955372, 13.7814035391, 14.0809753901, 14.6754076184