Dirichlet series
| L(s) = 1 | − 3-s + 4-s − 2·7-s + 6·11-s − 12-s + 4·13-s + 16-s − 2·17-s − 8·19-s + 2·21-s − 6·23-s − 10·25-s + 4·27-s − 2·28-s − 14·31-s − 6·33-s + 4·37-s − 4·39-s + 12·41-s + 4·43-s + 6·44-s + 12·47-s − 48-s − 2·49-s + 2·51-s + 4·52-s + 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.755·7-s + 1.80·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 1.83·19-s + 0.436·21-s − 1.25·23-s − 2·25-s + 0.769·27-s − 0.377·28-s − 2.51·31-s − 1.04·33-s + 0.657·37-s − 0.640·39-s + 1.87·41-s + 0.609·43-s + 0.904·44-s + 1.75·47-s − 0.144·48-s − 2/7·49-s + 0.280·51-s + 0.554·52-s + 1.05·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 3468 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3468 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(3468\) = \(2^{2} \cdot 3 \cdot 17^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.221122\) |
| Root analytic conductor: | \(0.685738\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 3468,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6958450097\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6958450097\) |
| \(L(\frac{3}{2})\) | 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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | ||
| 17 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.5.a_k |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.c_g | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ag_w | |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.13.ae_be | |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.19.i_cc | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.g_bu | |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.29.a_cg | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.31.o_dy | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.37.ae_bq | |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.41.am_eo | |
| 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$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.am_dq | |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.a_cs | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.m_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.61.ae_dm | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.ae_dy | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.71.ag_fm | |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.73.ae_fu | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.79.c_da | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.83.am_gk | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) | 2.89.y_la | |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) | 2.97.abc_pa | |
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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
−17.9303168931, −17.4469801976, −16.8465017077, −16.6317948443, −15.8238007055, −15.7312805996, −14.8984714428, −14.2197849424, −13.9862864899, −12.8595646329, −12.7863839659, −11.9583441737, −11.5065529704, −10.8489466872, −10.5768998024, −9.36174658707, −9.28607198826, −8.36750860338, −7.50641789037, −6.65330731261, −5.99911855172, −5.96956680213, −4.05372696694, −3.90229547123, −2.04126125287, 2.04126125287, 3.90229547123, 4.05372696694, 5.96956680213, 5.99911855172, 6.65330731261, 7.50641789037, 8.36750860338, 9.28607198826, 9.36174658707, 10.5768998024, 10.8489466872, 11.5065529704, 11.9583441737, 12.7863839659, 12.8595646329, 13.9862864899, 14.2197849424, 14.8984714428, 15.7312805996, 15.8238007055, 16.6317948443, 16.8465017077, 17.4469801976, 17.9303168931