Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 3·8-s + 3·9-s − 4·11-s + 2·12-s − 4·13-s − 16-s + 8·17-s − 3·18-s + 4·22-s − 8·23-s − 6·24-s + 25-s + 4·26-s − 4·27-s − 4·31-s − 5·32-s + 8·33-s − 8·34-s − 3·36-s − 20·37-s + 8·39-s + 12·41-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.06·8-s + 9-s − 1.20·11-s + 0.577·12-s − 1.10·13-s − 1/4·16-s + 1.94·17-s − 0.707·18-s + 0.852·22-s − 1.66·23-s − 1.22·24-s + 1/5·25-s + 0.784·26-s − 0.769·27-s − 0.718·31-s − 0.883·32-s + 1.39·33-s − 1.37·34-s − 1/2·36-s − 3.28·37-s + 1.28·39-s + 1.87·41-s + 0.603·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(14400\) = \(2^{6} \cdot 3^{2} \cdot 5^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.918156\) |
| Root analytic conductor: | \(0.978879\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 14400,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.7.a_o |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.e_w | |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.13.e_be | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.17.ai_bu | |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.a_cc | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.31.e_ck | |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) | 2.37.u_gs | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.41.am_dy | |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.a_cs | |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.47.a_be | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.53.i_di | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.59.ae_di | |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.61.a_eo | |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.a_ak | |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) | 2.71.q_hy | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.73.e_g | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.79.am_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$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.89.u_kc | |
| 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
−16.3291719014, −15.9668972191, −15.9590260302, −14.8569274509, −14.4459696363, −14.0069258505, −13.4321781833, −12.6461787614, −12.4649334276, −12.0745174066, −11.3233740297, −10.6789224512, −10.2832461615, −9.93128976176, −9.52345167581, −8.67148330402, −7.88407908599, −7.66488013442, −7.08681504005, −6.11496655716, −5.27165295209, −5.23920392625, −4.28889632981, −3.29264628325, −1.75820205156, 0, 1.75820205156, 3.29264628325, 4.28889632981, 5.23920392625, 5.27165295209, 6.11496655716, 7.08681504005, 7.66488013442, 7.88407908599, 8.67148330402, 9.52345167581, 9.93128976176, 10.2832461615, 10.6789224512, 11.3233740297, 12.0745174066, 12.4649334276, 12.6461787614, 13.4321781833, 14.0069258505, 14.4459696363, 14.8569274509, 15.9590260302, 15.9668972191, 16.3291719014