Dirichlet series
| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 4·5-s + 2·6-s − 4·7-s − 4·8-s + 8·10-s − 4·11-s − 2·12-s − 4·13-s + 8·14-s + 4·15-s + 8·16-s − 5·19-s − 8·20-s + 4·21-s + 8·22-s + 9·23-s + 4·24-s + 4·25-s + 8·26-s − 2·27-s − 8·28-s − 3·29-s − 8·30-s − 9·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s − 1.51·7-s − 1.41·8-s + 2.52·10-s − 1.20·11-s − 0.577·12-s − 1.10·13-s + 2.13·14-s + 1.03·15-s + 2·16-s − 1.14·19-s − 1.78·20-s + 0.872·21-s + 1.70·22-s + 1.87·23-s + 0.816·24-s + 4/5·25-s + 1.56·26-s − 0.384·27-s − 1.51·28-s − 0.557·29-s − 1.46·30-s − 1.61·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 22931 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22931 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(22931\) = \(23 \cdot 997\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.46210\) |
| Root analytic conductor: | \(1.09962\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 22931,\ (\ :1/2, 1/2),\ 1)\) |
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 | 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 8 T + p T^{2} ) \) | |
| 997 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 14 T + p T^{2} ) \) | ||
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.2.c_c |
| 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_b | |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.5.e_m | |
| 7 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.7.e_m | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.e_k | |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.13.e_s | |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) | 2.17.a_ae | |
| 19 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.19.f_p | |
| 29 | $D_{4}$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.29.d_ai | |
| 31 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.31.j_bu | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.37.d_cw | |
| 41 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.41.d_bo | |
| 43 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.43.c_ba | |
| 47 | $D_{4}$ | \( 1 + 7 T + 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.47.h_w | |
| 53 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.53.am_fa | |
| 59 | $D_{4}$ | \( 1 - 11 T + 141 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.59.al_fl | |
| 61 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.61.ag_w | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.r_hm | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.71.ah_fe | |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) | 2.73.a_cw | |
| 79 | $D_{4}$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.79.b_abq | |
| 83 | $D_{4}$ | \( 1 - 7 T + 114 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.83.ah_ek | |
| 89 | $D_{4}$ | \( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.89.e_es | |
| 97 | $D_{4}$ | \( 1 + 13 T + p T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.97.n_dt | |
| show more | ||||
| show less | ||||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.2820618579, −15.7269051944, −15.1854144340, −14.9487566307, −14.7368031296, −13.2630607978, −13.1058324262, −12.5600134843, −12.1481311270, −11.6734638759, −11.1824910939, −10.6643585748, −10.1865510569, −9.62910838679, −9.17569241079, −8.61159397595, −8.10301305481, −7.53211803717, −7.01167769494, −6.64494745331, −5.63231227565, −5.18309423523, −3.98517339339, −3.34236005048, −2.59754431618, 0, 0, 2.59754431618, 3.34236005048, 3.98517339339, 5.18309423523, 5.63231227565, 6.64494745331, 7.01167769494, 7.53211803717, 8.10301305481, 8.61159397595, 9.17569241079, 9.62910838679, 10.1865510569, 10.6643585748, 11.1824910939, 11.6734638759, 12.1481311270, 12.5600134843, 13.1058324262, 13.2630607978, 14.7368031296, 14.9487566307, 15.1854144340, 15.7269051944, 16.2820618579