Dirichlet series
| L(s) = 1 | − 3-s + 4-s − 2·5-s − 3·7-s − 12-s + 13-s + 2·15-s + 16-s + 3·19-s − 2·20-s + 3·21-s − 2·23-s + 3·25-s + 27-s − 3·28-s − 6·31-s + 6·35-s + 3·37-s − 39-s + 14·41-s − 10·43-s + 2·47-s − 48-s + 5·49-s + 52-s − 4·53-s − 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.894·5-s − 1.13·7-s − 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s + 0.688·19-s − 0.447·20-s + 0.654·21-s − 0.417·23-s + 3/5·25-s + 0.192·27-s − 0.566·28-s − 1.07·31-s + 1.01·35-s + 0.493·37-s − 0.160·39-s + 2.18·41-s − 1.52·43-s + 0.291·47-s − 0.144·48-s + 5/7·49-s + 0.138·52-s − 0.549·53-s − 0.397·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1116\) = \(2^{2} \cdot 3^{2} \cdot 31\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0711571\) |
| Root analytic conductor: | \(0.516481\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 1116,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4354897184\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4354897184\) |
| \(L(\frac{3}{2})\) | 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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 3 | $C_2$ | \( 1 + T + T^{2} \) | ||
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) | ||
| good | 5 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_b |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.7.d_e | |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) | 2.11.a_af | |
| 13 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) | 2.13.ab_a | |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) | 2.17.a_w | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.ad_k | |
| 23 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.23.c_h | |
| 29 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) | 2.29.a_q | |
| 37 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_ai | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.41.ao_el | |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.43.k_by | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.47.ac_cs | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.e_dq | |
| 59 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.59.ak_de | |
| 61 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.61.d_cs | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.67.c_cc | |
| 71 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.71.m_dw | |
| 73 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.73.b_aw | |
| 79 | $D_{4}$ | \( 1 + 7 T + 62 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.79.h_ck | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.83.ac_dz | |
| 89 | $D_{4}$ | \( 1 + 8 T + 16 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.89.i_q | |
| 97 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.97.g_fm | |
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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
−19.8269873343, −19.2883757539, −18.8516756234, −18.0948767073, −17.7136828040, −16.6192987778, −16.5175014031, −15.9517998857, −15.5242386030, −14.7983059252, −14.1647667290, −13.2277394012, −12.7370138563, −12.1259293035, −11.4968156485, −11.0450555869, −10.2451872085, −9.55019977604, −8.71792977509, −7.72875675432, −7.10805164474, −6.27604807303, −5.52797435783, −4.15236732343, −3.09412679935, 3.09412679935, 4.15236732343, 5.52797435783, 6.27604807303, 7.10805164474, 7.72875675432, 8.71792977509, 9.55019977604, 10.2451872085, 11.0450555869, 11.4968156485, 12.1259293035, 12.7370138563, 13.2277394012, 14.1647667290, 14.7983059252, 15.5242386030, 15.9517998857, 16.5175014031, 16.6192987778, 17.7136828040, 18.0948767073, 18.8516756234, 19.2883757539, 19.8269873343