Dirichlet series
| L(s) = 1 | − 2·3-s + 4-s − 5-s − 2·9-s + 4·11-s − 2·12-s − 10·13-s + 2·15-s + 16-s + 8·17-s + 2·19-s − 20-s − 4·25-s + 10·27-s + 4·31-s − 8·33-s − 2·36-s − 8·37-s + 20·39-s + 8·41-s + 12·43-s + 4·44-s + 2·45-s − 4·47-s − 2·48-s + 49-s − 16·51-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 0.447·5-s − 2/3·9-s + 1.20·11-s − 0.577·12-s − 2.77·13-s + 0.516·15-s + 1/4·16-s + 1.94·17-s + 0.458·19-s − 0.223·20-s − 4/5·25-s + 1.92·27-s + 0.718·31-s − 1.39·33-s − 1/3·36-s − 1.31·37-s + 3.20·39-s + 1.24·41-s + 1.82·43-s + 0.603·44-s + 0.298·45-s − 0.583·47-s − 0.288·48-s + 1/7·49-s − 2.24·51-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(980\) = \(2^{2} \cdot 5 \cdot 7^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0624856\) |
| Root analytic conductor: | \(0.499971\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 980,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3897644204\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3897644204\) |
| \(L(\frac{3}{2})\) | 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$ | \( ( 1 - T )( 1 + T ) \) | |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.c_g |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ae_w | |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.k_by | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.17.ai_bu | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.19.ac_bm | |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.23.a_bu | |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.29.a_w | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.31.ae_be | |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.41.ai_dq | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.43.am_eo | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.47.e_ac | |
| 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$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.59.o_gk | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.61.g_k | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.q_ha | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.71.q_fm | |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.73.ae_fu | |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.79.a_dq | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.83.ac_eo | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.89.ae_eo | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.97.i_gs | |
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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
−19.5800270198, −19.5318859919, −19.3881993684, −18.2120541317, −17.4014938000, −17.2128532162, −16.8198467621, −16.3370810014, −15.5600040575, −14.6077730657, −14.5358664373, −13.8941777274, −12.3052609901, −12.1470736931, −11.9657271713, −11.2313614144, −10.3759943109, −9.76554711946, −8.97764609237, −7.57571100089, −7.45420248730, −6.17616144092, −5.57928681743, −4.63923341950, −3.00381372509, 3.00381372509, 4.63923341950, 5.57928681743, 6.17616144092, 7.45420248730, 7.57571100089, 8.97764609237, 9.76554711946, 10.3759943109, 11.2313614144, 11.9657271713, 12.1470736931, 12.3052609901, 13.8941777274, 14.5358664373, 14.6077730657, 15.5600040575, 16.3370810014, 16.8198467621, 17.2128532162, 17.4014938000, 18.2120541317, 19.3881993684, 19.5318859919, 19.5800270198