Dirichlet series
| L(s) = 1 | − 3-s + 2·5-s + 5·7-s − 9-s + 13-s − 2·15-s − 17-s + 2·19-s − 5·21-s − 2·23-s − 2·25-s + 6·29-s + 8·31-s + 10·35-s + 3·37-s − 39-s + 4·41-s − 3·43-s − 2·45-s + 17·47-s + 11·49-s + 51-s − 2·57-s − 8·61-s − 5·63-s + 2·65-s − 6·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.88·7-s − 1/3·9-s + 0.277·13-s − 0.516·15-s − 0.242·17-s + 0.458·19-s − 1.09·21-s − 0.417·23-s − 2/5·25-s + 1.11·29-s + 1.43·31-s + 1.69·35-s + 0.493·37-s − 0.160·39-s + 0.624·41-s − 0.457·43-s − 0.298·45-s + 2.47·47-s + 11/7·49-s + 0.140·51-s − 0.264·57-s − 1.02·61-s − 0.629·63-s + 0.248·65-s − 0.733·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(266240\) = \(2^{12} \cdot 5 \cdot 13\) |
| Sign: | $1$ |
| Analytic conductor: | \(16.9756\) |
| Root analytic conductor: | \(2.02981\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 266240,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.191897152\) |
| \(L(\frac12)\) | \(\approx\) | \(2.191897152\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_c |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) | 2.7.af_o | |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | 2.11.a_ak | |
| 17 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.17.b_am | |
| 19 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.19.ac_ac | |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.23.c_c | |
| 29 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.29.ag_ba | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.ai_ck | |
| 37 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_y | |
| 41 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.41.ae_o | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.43.d_cg | |
| 47 | $D_{4}$ | \( 1 - 17 T + 150 T^{2} - 17 p T^{3} + p^{2} T^{4} \) | 2.47.ar_fu | |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) | 2.53.a_cg | |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) | 2.59.a_da | |
| 61 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.61.i_eo | |
| 67 | $D_{4}$ | \( 1 + 6 T - 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.67.g_as | |
| 71 | $D_{4}$ | \( 1 + 5 T + 70 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.71.f_cs | |
| 73 | $D_{4}$ | \( 1 + 14 T + 182 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.73.o_ha | |
| 79 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.79.am_ew | |
| 83 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.83.e_eo | |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.89.a_fm | |
| 97 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.97.k_eg | |
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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
−13.4651626106, −12.6427595990, −12.1524318615, −11.8834781530, −11.5313875304, −11.0921554864, −10.7321792050, −10.3025812151, −9.89259415219, −9.36805688154, −8.76024740627, −8.51876796070, −7.92481740269, −7.62321373344, −7.03509223639, −6.32443275062, −5.89144983650, −5.62342607307, −5.05211378432, −4.35537759119, −4.31199742931, −3.06312044869, −2.42658036327, −1.72758603344, −1.00588345512, 1.00588345512, 1.72758603344, 2.42658036327, 3.06312044869, 4.31199742931, 4.35537759119, 5.05211378432, 5.62342607307, 5.89144983650, 6.32443275062, 7.03509223639, 7.62321373344, 7.92481740269, 8.51876796070, 8.76024740627, 9.36805688154, 9.89259415219, 10.3025812151, 10.7321792050, 11.0921554864, 11.5313875304, 11.8834781530, 12.1524318615, 12.6427595990, 13.4651626106