Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s − 4-s − 5-s + 2·6-s + 3·8-s + 3·9-s + 10-s + 2·12-s − 4·13-s + 2·15-s − 16-s + 4·17-s − 3·18-s + 20-s − 8·23-s − 6·24-s − 2·25-s + 4·26-s − 4·27-s + 4·29-s − 2·30-s + 8·31-s − 5·32-s − 4·34-s − 3·36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s + 1.06·8-s + 9-s + 0.316·10-s + 0.577·12-s − 1.10·13-s + 0.516·15-s − 1/4·16-s + 0.970·17-s − 0.707·18-s + 0.223·20-s − 1.66·23-s − 1.22·24-s − 2/5·25-s + 0.784·26-s − 0.769·27-s + 0.742·29-s − 0.365·30-s + 1.43·31-s − 0.883·32-s − 0.685·34-s − 1/2·36-s − 0.657·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(360\) = \(2^{3} \cdot 3^{2} \cdot 5\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0229539\) |
| Root analytic conductor: | \(0.389237\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 360,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1887763985\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1887763985\) |
| \(L(\frac{3}{2})\) | 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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | ||
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.7.a_o |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.a_g | |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.13.e_be | |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.17.ae_bm | |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.a_w | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.i_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.ae_bu | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.ai_ck | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.e_o | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.ae_w | |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.43.ai_dy | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.ai_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.53.m_ew | |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.59.a_dy | |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.61.e_ew | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.ai_di | |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.a_da | |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) | 2.73.au_jm | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.79.i_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.83.ai_eo | |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.89.m_ig | |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.97.ae_hq | |
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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
−19.8934316645, −19.1551537949, −19.1104624590, −18.1058537571, −17.6707519500, −17.2359534775, −16.6272610009, −15.9590260302, −15.4732516475, −14.1819882616, −14.0069258505, −12.6461787614, −12.3172568607, −11.5820474614, −10.6789224512, −9.96467191573, −9.52345167581, −8.09899069409, −7.66488013442, −6.42897107073, −5.23920392625, −4.25303028693, 4.25303028693, 5.23920392625, 6.42897107073, 7.66488013442, 8.09899069409, 9.52345167581, 9.96467191573, 10.6789224512, 11.5820474614, 12.3172568607, 12.6461787614, 14.0069258505, 14.1819882616, 15.4732516475, 15.9590260302, 16.6272610009, 17.2359534775, 17.6707519500, 18.1058537571, 19.1104624590, 19.1551537949, 19.8934316645