Dirichlet series
| L(s) = 1 | − 3-s + 2·5-s + 9-s − 4·11-s − 2·15-s + 2·17-s − 4·19-s + 2·25-s − 27-s + 10·29-s + 8·31-s + 4·33-s − 4·37-s − 6·41-s + 4·43-s + 2·45-s + 8·47-s + 2·49-s − 2·51-s + 2·53-s − 8·55-s + 4·57-s + 4·59-s − 4·61-s − 4·67-s + 16·71-s − 2·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.516·15-s + 0.485·17-s − 0.917·19-s + 2/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.696·33-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 2/7·49-s − 0.280·51-s + 0.274·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.512·61-s − 0.488·67-s + 1.89·71-s − 0.230·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(110592\) = \(2^{12} \cdot 3^{3}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.05144\) |
| Root analytic conductor: | \(1.62955\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 110592,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.387976757\) |
| \(L(\frac12)\) | \(\approx\) | \(1.387976757\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 3 | $C_1$ | \( 1 + T \) | ||
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.ac_c |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.a_ac | |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.11.e_g | |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) | 2.13.a_g | |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.17.ac_s | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.19.e_g | |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.23.a_ac | |
| 29 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.29.ak_co | |
| 31 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.31.ai_bu | |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.37.e_ac | |
| 41 | $D_{4}$ | \( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.41.g_c | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.ae_cc | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.47.ai_bu | |
| 53 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.53.ac_by | |
| 59 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.59.ae_ak | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.61.e_as | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.67.e_acg | |
| 71 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) | 2.71.aq_gc | |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.73.a_bu | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.79.ai_eg | |
| 83 | $D_{4}$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) | 2.83.au_ig | |
| 89 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.89.g_fq | |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) | 2.97.a_ck | |
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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
−14.0061293475, −13.4369643993, −13.2456686316, −12.5332555140, −12.2100256290, −12.0127678007, −11.2304506816, −10.6612128358, −10.4765159141, −10.0870875823, −9.68293166700, −9.01505965855, −8.46452091313, −8.11445027001, −7.51743354768, −6.84922420430, −6.40213341851, −5.99442248722, −5.34626481547, −4.93672789073, −4.42311929675, −3.50485885256, −2.64964239777, −2.14027662346, −0.896963152278, 0.896963152278, 2.14027662346, 2.64964239777, 3.50485885256, 4.42311929675, 4.93672789073, 5.34626481547, 5.99442248722, 6.40213341851, 6.84922420430, 7.51743354768, 8.11445027001, 8.46452091313, 9.01505965855, 9.68293166700, 10.0870875823, 10.4765159141, 10.6612128358, 11.2304506816, 12.0127678007, 12.2100256290, 12.5332555140, 13.2456686316, 13.4369643993, 14.0061293475