Dirichlet series
| L(s) = 1 | − 3-s − 7-s − 2·9-s − 4·17-s + 4·19-s + 21-s − 8·23-s − 6·25-s + 2·27-s + 12·29-s + 16·31-s + 4·37-s − 4·41-s − 8·47-s − 6·49-s + 4·51-s + 4·53-s − 4·57-s + 4·59-s − 8·61-s + 2·63-s − 8·67-s + 8·69-s + 20·73-s + 6·75-s + 8·79-s + 7·81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.970·17-s + 0.917·19-s + 0.218·21-s − 1.66·23-s − 6/5·25-s + 0.384·27-s + 2.22·29-s + 2.87·31-s + 0.657·37-s − 0.624·41-s − 1.16·47-s − 6/7·49-s + 0.560·51-s + 0.549·53-s − 0.529·57-s + 0.520·59-s − 1.02·61-s + 0.251·63-s − 0.977·67-s + 0.963·69-s + 2.34·73-s + 0.692·75-s + 0.900·79-s + 7/9·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1344\) = \(2^{6} \cdot 3 \cdot 7\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0856946\) |
| Root analytic conductor: | \(0.541051\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 1344,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4714942810\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4714942810\) |
| \(L(\frac{3}{2})\) | 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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.a_g |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.a_g | |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.13.a_w | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.e_w | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.ae_g | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.i_bu | |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.29.am_dq | |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | 2.31.aq_ew | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.37.ae_ck | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.e_cs | |
| 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$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.47.i_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.53.ae_dq | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.ae_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.61.i_fe | |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.67.i_fu | |
| 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$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.79.ai_be | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.83.ae_fe | |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.89.m_ig | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.97.e_ha | |
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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
−19.5501882732, −19.1551537949, −18.0743360042, −17.6857167880, −17.6707519500, −16.6272610009, −16.2852315391, −15.4732516475, −15.4132338247, −14.1819882616, −13.7774671320, −13.4509485366, −12.3172568607, −11.8414629768, −11.5820474614, −10.6022424461, −9.96467191573, −9.45734931342, −8.18842041768, −8.09899069409, −6.42897107073, −6.42262084814, −5.22316409435, −4.25303028693, −2.79183800613, 2.79183800613, 4.25303028693, 5.22316409435, 6.42262084814, 6.42897107073, 8.09899069409, 8.18842041768, 9.45734931342, 9.96467191573, 10.6022424461, 11.5820474614, 11.8414629768, 12.3172568607, 13.4509485366, 13.7774671320, 14.1819882616, 15.4132338247, 15.4732516475, 16.2852315391, 16.6272610009, 17.6707519500, 17.6857167880, 18.0743360042, 19.1551537949, 19.5501882732