Dirichlet series
| L(s) = 1 | − 2·2-s − 4·3-s + 3·4-s + 8·6-s + 2·7-s − 4·8-s + 6·9-s − 12·12-s − 8·13-s − 4·14-s + 5·16-s + 12·17-s − 12·18-s + 4·19-s − 8·21-s + 16·24-s − 10·25-s + 16·26-s + 4·27-s + 6·28-s − 12·29-s − 8·31-s − 6·32-s − 24·34-s + 18·36-s + 4·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 3/2·4-s + 3.26·6-s + 0.755·7-s − 1.41·8-s + 2·9-s − 3.46·12-s − 2.21·13-s − 1.06·14-s + 5/4·16-s + 2.91·17-s − 2.82·18-s + 0.917·19-s − 1.74·21-s + 3.26·24-s − 2·25-s + 3.13·26-s + 0.769·27-s + 1.13·28-s − 2.22·29-s − 1.43·31-s − 1.06·32-s − 4.11·34-s + 3·36-s + 0.657·37-s − 1.29·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(196\) = \(2^{2} \cdot 7^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0124971\) |
| Root analytic conductor: | \(0.334350\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 196,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1090476651\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1090476651\) |
| \(L(\frac{3}{2})\) | 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 | $C_1$ | \( ( 1 + T )^{2} \) | |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) | ||
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.3.e_k |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.5.a_k | |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.11.a_w | |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.13.i_bq | |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.17.am_cs | |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.19.ae_bq | |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.23.a_bu | |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.29.m_dq | |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.31.i_da | |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.37.ae_da | |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.41.am_eo | |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | 2.43.aq_fu | |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) | 2.47.y_je | |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.53.am_fm | |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.59.m_fy | |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | 2.61.aq_he | |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.67.i_fu | |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.71.a_fm | |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.73.ae_fu | |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | 2.79.aq_io | |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.83.m_hu | |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.89.m_ig | |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) | 2.97.u_li | |
| show more | ||||
| show less | ||||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.5800270198, −19.5800270198, −18.2120541317, −18.2120541317, −17.2128532162, −17.2128532162, −16.3370810014, −16.3370810014, −14.6077730657, −14.6077730657, −12.3052609901, −12.3052609901, −11.2313614144, −11.2313614144, −9.76554711946, −9.76554711946, −7.57571100089, −7.57571100089, −5.57928681743, −5.57928681743, 5.57928681743, 5.57928681743, 7.57571100089, 7.57571100089, 9.76554711946, 9.76554711946, 11.2313614144, 11.2313614144, 12.3052609901, 12.3052609901, 14.6077730657, 14.6077730657, 16.3370810014, 16.3370810014, 17.2128532162, 17.2128532162, 18.2120541317, 18.2120541317, 19.5800270198, 19.5800270198