Dirichlet series
| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s − 4·8-s + 2·9-s + 2·10-s + 11-s − 2·12-s + 2·13-s + 4·14-s + 15-s + 8·16-s + 3·17-s − 4·18-s − 2·20-s + 2·21-s − 2·22-s − 3·23-s + 4·24-s − 6·25-s − 4·26-s − 6·27-s − 4·28-s + 29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s + 2·16-s + 0.727·17-s − 0.942·18-s − 0.447·20-s + 0.436·21-s − 0.426·22-s − 0.625·23-s + 0.816·24-s − 6/5·25-s − 0.784·26-s − 1.15·27-s − 0.755·28-s + 0.185·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(349\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0222525\) |
| Root analytic conductor: | \(0.386229\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 349,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1656123320\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1656123320\) |
| \(L(\frac{3}{2})\) | 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 | 349 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 12 T + p T^{2} ) \) | |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.2.c_c |
| 3 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_ab | |
| 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.5.b_h | |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_m | |
| 11 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.11.ab_ag | |
| 13 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.13.ac_o | |
| 17 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.17.ad_l | |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.19.a_c | |
| 23 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.23.d_w | |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.29.ab_u | |
| 31 | $D_{4}$ | \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.31.ab_bt | |
| 37 | $D_{4}$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_f | |
| 41 | $D_{4}$ | \( 1 + T - 60 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.41.b_aci | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.43.e_cc | |
| 47 | $D_{4}$ | \( 1 - 6 T + 80 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.47.ag_dc | |
| 53 | $D_{4}$ | \( 1 - 9 T + 42 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.53.aj_bq | |
| 59 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.59.k_da | |
| 61 | $D_{4}$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.61.ab_acu | |
| 67 | $D_{4}$ | \( 1 - 11 T + p T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.67.al_cp | |
| 71 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) | 2.71.a_dc | |
| 73 | $D_{4}$ | \( 1 - 11 T + 112 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.73.al_ei | |
| 79 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.79.c_u | |
| 83 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.83.ac_dm | |
| 89 | $D_{4}$ | \( 1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.89.ae_ee | |
| 97 | $D_{4}$ | \( 1 + T + 86 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.97.b_di | |
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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
−19.8801206441, −19.4683267789, −18.6964414228, −18.5341976818, −17.8121239507, −17.1869788794, −16.6930224052, −15.9057621792, −15.5783517056, −14.8149750036, −13.7491444561, −12.8379449443, −12.0502441033, −11.6421604602, −10.6589739621, −9.76213624041, −9.51285691832, −8.39435762732, −7.68529649455, −6.54603331915, −5.77378692411, −3.69794085043, 3.69794085043, 5.77378692411, 6.54603331915, 7.68529649455, 8.39435762732, 9.51285691832, 9.76213624041, 10.6589739621, 11.6421604602, 12.0502441033, 12.8379449443, 13.7491444561, 14.8149750036, 15.5783517056, 15.9057621792, 16.6930224052, 17.1869788794, 17.8121239507, 18.5341976818, 18.6964414228, 19.4683267789, 19.8801206441