Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 3·8-s + 3·9-s − 8·11-s + 2·12-s − 16-s − 3·18-s − 4·19-s + 8·22-s − 4·23-s − 6·24-s + 25-s − 4·27-s − 8·29-s − 5·32-s + 16·33-s − 3·36-s − 8·37-s + 4·38-s + 4·41-s + 8·44-s + 4·46-s + 20·47-s + 2·48-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 − 2.41·11-s + 0.577·12-s − 1/4·16-s − 0.707·18-s − 0.917·19-s + 1.70·22-s − 0.834·23-s − 1.22·24-s + 1/5·25-s − 0.769·27-s − 1.48·29-s − 0.883·32-s + 2.78·33-s − 1/2·36-s − 1.31·37-s + 0.648·38-s + 0.624·41-s + 1.20·44-s + 0.589·46-s + 2.91·47-s + 0.288·48-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(7200\) = \(2^{5} \cdot 3^{2} \cdot 5^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.459078\) |
| Root analytic conductor: | \(0.823136\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 7200,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.11.i_bm | |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.13.a_w | |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.17.a_be | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.19.e_g | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.23.e_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.29.i_cs | |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.31.a_ck | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.i_cc | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.ae_w | |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.a_cs | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) | 2.47.au_hi | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.53.q_gk | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.q_gk | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.61.am_dq | |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) | 2.67.ay_ks | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.i_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.73.am_gk | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.79.ai_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.83.aq_ig | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.89.e_gk | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.97.m_gk | |
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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
−17.2359534775, −17.0285068022, −16.2652641967, −15.9590260302, −15.5114390469, −14.9757145558, −14.0069258505, −13.7392025398, −12.9418318979, −12.6461787614, −12.3574631320, −11.1766293089, −10.7974752774, −10.6789224512, −9.83549734579, −9.52345167581, −8.58268357776, −7.88889261724, −7.66488013442, −6.76366111245, −5.86487938441, −5.23920392625, −4.78210199118, −3.76484405046, −2.15654793578, 0, 2.15654793578, 3.76484405046, 4.78210199118, 5.23920392625, 5.86487938441, 6.76366111245, 7.66488013442, 7.88889261724, 8.58268357776, 9.52345167581, 9.83549734579, 10.6789224512, 10.7974752774, 11.1766293089, 12.3574631320, 12.6461787614, 12.9418318979, 13.7392025398, 14.0069258505, 14.9757145558, 15.5114390469, 15.9590260302, 16.2652641967, 17.0285068022, 17.2359534775