Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 8-s − 2·9-s − 2·10-s − 4·11-s − 2·12-s − 2·13-s − 4·15-s + 16-s + 2·18-s + 10·19-s + 2·20-s + 4·22-s + 2·24-s − 6·25-s + 2·26-s + 10·27-s + 4·30-s + 4·31-s − 32-s + 8·33-s − 2·36-s − 10·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.353·8-s − 2/3·9-s − 0.632·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s − 1.03·15-s + 1/4·16-s + 0.471·18-s + 2.29·19-s + 0.447·20-s + 0.852·22-s + 0.408·24-s − 6/5·25-s + 0.392·26-s + 1.92·27-s + 0.730·30-s + 0.718·31-s − 0.176·32-s + 1.39·33-s − 1/3·36-s − 1.62·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(784\) = \(2^{4} \cdot 7^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0499885\) |
| Root analytic conductor: | \(0.472843\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 784,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2887965445\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2887965445\) |
| \(L(\frac{3}{2})\) | 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 | $C_1$ | \( 1 + T \) | |
| 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 |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.5.ac_k | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.e_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.13.c_s | |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.a_ac | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.19.ak_cc | |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.37.a_cs | |
| 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.ae_cc | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.47.u_hi | |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.53.am_fm | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.59.g_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.61.ac_cw | |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.67.i_fu | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.i_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.73.am_gk | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) | 2.79.ay_la | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.83.ac_eo | |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.89.m_ig | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.97.q_ju | |
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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, −19.5501882732, −18.2120541317, −18.0743360042, −17.6857167880, −17.2128532162, −16.3370810014, −16.2852315391, −15.4132338247, −14.6077730657, −13.7774671320, −13.4509485366, −12.3052609901, −11.8414629768, −11.2313614144, −10.6022424461, −9.76554711946, −9.45734931342, −8.18842041768, −7.57571100089, −6.42262084814, −5.57928681743, −5.22316409435, −2.79183800613, 2.79183800613, 5.22316409435, 5.57928681743, 6.42262084814, 7.57571100089, 8.18842041768, 9.45734931342, 9.76554711946, 10.6022424461, 11.2313614144, 11.8414629768, 12.3052609901, 13.4509485366, 13.7774671320, 14.6077730657, 15.4132338247, 16.2852315391, 16.3370810014, 17.2128532162, 17.6857167880, 18.0743360042, 18.2120541317, 19.5501882732, 19.5800270198