Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s − 2·4-s − 5-s + 2·6-s − 4·7-s + 3·8-s − 2·9-s + 10-s − 4·11-s + 4·12-s + 4·14-s + 2·15-s + 16-s − 3·17-s + 2·18-s − 2·19-s + 2·20-s + 8·21-s + 4·22-s − 8·23-s − 6·24-s − 2·25-s + 10·27-s + 8·28-s + 2·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s − 0.447·5-s + 0.816·6-s − 1.51·7-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 1.20·11-s + 1.15·12-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.458·19-s + 0.447·20-s + 1.74·21-s + 0.852·22-s − 1.66·23-s − 1.22·24-s − 2/5·25-s + 1.92·27-s + 1.51·28-s + 0.371·29-s − 0.365·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 25505 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25505 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(25505\) = \(5 \cdot 5101\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.62622\) |
| Root analytic conductor: | \(1.12926\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 25505,\ (\ :1/2, 1/2),\ 1)\) |
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | |
| 5101 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 100 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.2.b_d |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.c_g | |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.7.e_k | |
| 11 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.11.e_q | |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) | 2.13.a_f | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.19.c_j | |
| 23 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.23.i_cc | |
| 29 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.29.ac_c | |
| 31 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.31.c_y | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.37.h_ce | |
| 41 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.41.d_bg | |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) | 2.43.a_cc | |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) | 2.47.a_bo | |
| 53 | $D_{4}$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.53.d_abm | |
| 59 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) | 2.59.a_b | |
| 61 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.61.b_am | |
| 67 | $D_{4}$ | \( 1 - 12 T + 162 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.67.am_gg | |
| 71 | $D_{4}$ | \( 1 + 4 T + 119 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.71.e_ep | |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) | 2.73.a_da | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.79.i_be | |
| 83 | $D_{4}$ | \( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.83.ag_y | |
| 89 | $D_{4}$ | \( 1 - T + 90 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.89.ab_dm | |
| 97 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.97.ag_br | |
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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
−16.1475156548, −15.6223215723, −15.3289428372, −14.3791652013, −13.9858863690, −13.5456167194, −13.0052542786, −12.6571333158, −12.1138456394, −11.6351451476, −11.1020286684, −10.3608175279, −10.2967574353, −9.65024694024, −9.06099060593, −8.56129724764, −8.14107487650, −7.53699016433, −6.52989794824, −6.32098889879, −5.48966199480, −5.10356276190, −4.23052270469, −3.50977157183, −2.51439873182, 0, 0, 2.51439873182, 3.50977157183, 4.23052270469, 5.10356276190, 5.48966199480, 6.32098889879, 6.52989794824, 7.53699016433, 8.14107487650, 8.56129724764, 9.06099060593, 9.65024694024, 10.2967574353, 10.3608175279, 11.1020286684, 11.6351451476, 12.1138456394, 12.6571333158, 13.0052542786, 13.5456167194, 13.9858863690, 14.3791652013, 15.3289428372, 15.6223215723, 16.1475156548