Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s + 3·8-s − 10-s − 2·11-s + 2·12-s − 2·13-s − 2·15-s − 16-s − 20-s + 2·22-s − 3·23-s − 6·24-s − 25-s + 2·26-s + 2·27-s + 2·29-s + 2·30-s − 8·31-s − 5·32-s + 4·33-s + 4·39-s + 3·40-s + 41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s + 1.06·8-s − 0.316·10-s − 0.603·11-s + 0.577·12-s − 0.554·13-s − 0.516·15-s − 1/4·16-s − 0.223·20-s + 0.426·22-s − 0.625·23-s − 1.22·24-s − 1/5·25-s + 0.392·26-s + 0.384·27-s + 0.371·29-s + 0.365·30-s − 1.43·31-s − 0.883·32-s + 0.696·33-s + 0.640·39-s + 0.474·40-s + 0.156·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 13696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(13696\) = \(2^{7} \cdot 107\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.873268\) |
| Root analytic conductor: | \(0.966689\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 13696,\ (\ :1/2, 1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | 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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 107 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 18 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.3.c_e |
| 5 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.5.ab_c | |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) | 2.7.a_g | |
| 11 | $D_{4}$ | \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.11.c_ae | |
| 13 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_m | |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.17.a_s | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.23.d_be | |
| 29 | $D_{4}$ | \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.29.ac_bc | |
| 31 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.31.i_cc | |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.37.a_k | |
| 41 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.41.ab_y | |
| 43 | $D_{4}$ | \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.43.af_cy | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.47.n_fe | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.53.ag_co | |
| 59 | $D_{4}$ | \( 1 + 15 T + 120 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.59.p_eq | |
| 61 | $D_{4}$ | \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.61.c_ai | |
| 67 | $D_{4}$ | \( 1 + 11 T + 120 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.67.l_eq | |
| 71 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.71.i_dm | |
| 73 | $D_{4}$ | \( 1 + 14 T + 142 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.73.o_fm | |
| 79 | $D_{4}$ | \( 1 - 15 T + 182 T^{2} - 15 p T^{3} + p^{2} T^{4} \) | 2.79.ap_ha | |
| 83 | $D_{4}$ | \( 1 + 4 T - 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.83.e_abg | |
| 89 | $D_{4}$ | \( 1 - 5 T + 56 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.89.af_ce | |
| 97 | $D_{4}$ | \( 1 + 2 T - 138 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.97.c_afi | |
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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
−16.6162068399, −16.2492132048, −15.7327915767, −14.9109791878, −14.5416195737, −13.9941035949, −13.4069502222, −13.0431998121, −12.4385585461, −11.9284322030, −11.3795204980, −10.7421547062, −10.5196466136, −9.82275689320, −9.41822411544, −8.83195094036, −8.11827626897, −7.60977515743, −6.99617863236, −6.00027706123, −5.74193592342, −4.94230335329, −4.45014043738, −3.21707949243, −1.83695808658, 0, 1.83695808658, 3.21707949243, 4.45014043738, 4.94230335329, 5.74193592342, 6.00027706123, 6.99617863236, 7.60977515743, 8.11827626897, 8.83195094036, 9.41822411544, 9.82275689320, 10.5196466136, 10.7421547062, 11.3795204980, 11.9284322030, 12.4385585461, 13.0431998121, 13.4069502222, 13.9941035949, 14.5416195737, 14.9109791878, 15.7327915767, 16.2492132048, 16.6162068399