Dirichlet series
| L(s) = 1 | − 2-s − 3-s − 4·5-s + 6-s + 8-s + 4·10-s − 6·11-s + 4·15-s − 16-s − 3·17-s − 4·19-s + 6·22-s + 2·23-s − 24-s + 6·25-s + 4·27-s − 4·30-s + 6·31-s + 6·33-s + 3·34-s − 8·37-s + 4·38-s − 4·40-s − 2·41-s + 2·43-s − 2·46-s + 48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1.78·5-s + 0.408·6-s + 0.353·8-s + 1.26·10-s − 1.80·11-s + 1.03·15-s − 1/4·16-s − 0.727·17-s − 0.917·19-s + 1.27·22-s + 0.417·23-s − 0.204·24-s + 6/5·25-s + 0.769·27-s − 0.730·30-s + 1.07·31-s + 1.04·33-s + 0.514·34-s − 1.31·37-s + 0.648·38-s − 0.632·40-s − 0.312·41-s + 0.304·43-s − 0.294·46-s + 0.144·48-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(4404\) = \(2^{2} \cdot 3 \cdot 367\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.280802\) |
| Root analytic conductor: | \(0.727948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 4404,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + T + T^{2} \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | ||
| 367 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 11 T + p T^{2} ) \) | ||
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) | 2.7.a_b | |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.11.g_bc | |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.13.a_b | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.e_bm | |
| 23 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.23.ac_bi | |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) | 2.29.a_ai | |
| 31 | $D_{4}$ | \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.31.ag_bf | |
| 37 | $D_{4}$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.37.i_bh | |
| 41 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_ac | |
| 43 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.43.ac_u | |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) | 2.47.a_ao | |
| 53 | $D_{4}$ | \( 1 - 4 T - 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.53.ae_abj | |
| 59 | $D_{4}$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.59.am_dt | |
| 61 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.61.c_v | |
| 67 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.67.m_fa | |
| 71 | $D_{4}$ | \( 1 - T + 28 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.71.ab_bc | |
| 73 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.73.c_dy | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.79.ag_eo | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.83.e_cs | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.89.ab_fg | |
| 97 | $D_{4}$ | \( 1 + 7 T + 170 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.97.h_go | |
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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
−17.8711246895, −17.5764737553, −16.9573970355, −16.2375221506, −16.0214050298, −15.4750297493, −15.1827325052, −14.5236074740, −13.4086669750, −13.3399672987, −12.3316018722, −12.1400411356, −11.3848505828, −10.7976020317, −10.5802718526, −9.86885381281, −8.80868692002, −8.34767115572, −7.96099915315, −7.22878005173, −6.65256286206, −5.47140108795, −4.73479999191, −4.00861013867, −2.74450396699, 0, 2.74450396699, 4.00861013867, 4.73479999191, 5.47140108795, 6.65256286206, 7.22878005173, 7.96099915315, 8.34767115572, 8.80868692002, 9.86885381281, 10.5802718526, 10.7976020317, 11.3848505828, 12.1400411356, 12.3316018722, 13.3399672987, 13.4086669750, 14.5236074740, 15.1827325052, 15.4750297493, 16.0214050298, 16.2375221506, 16.9573970355, 17.5764737553, 17.8711246895