Dirichlet series
| L(s) = 1 | − 3-s − 2·4-s − 7-s + 9-s − 3·11-s + 2·12-s − 13-s + 7·17-s + 3·19-s + 21-s − 2·25-s − 4·27-s + 2·28-s + 7·29-s + 3·33-s − 2·36-s + 37-s + 39-s − 5·41-s − 3·43-s + 6·44-s − 8·47-s − 11·49-s − 7·51-s + 2·52-s + 7·53-s − 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 0.277·13-s + 1.69·17-s + 0.688·19-s + 0.218·21-s − 2/5·25-s − 0.769·27-s + 0.377·28-s + 1.29·29-s + 0.522·33-s − 1/3·36-s + 0.164·37-s + 0.160·39-s − 0.780·41-s − 0.457·43-s + 0.904·44-s − 1.16·47-s − 1.57·49-s − 0.980·51-s + 0.277·52-s + 0.961·53-s − 0.397·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 797 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 797 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(797\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0508174\) |
| Root analytic conductor: | \(0.474791\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 797,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3559385607\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3559385607\) |
| \(L(\frac{3}{2})\) | 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 | 797 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 14 T + p T^{2} ) \) | |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | 2.2.a_c |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.b_a | |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.5.a_c | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.7.b_m | |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.11.d_k | |
| 13 | $D_{4}$ | \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_aj | |
| 17 | $D_{4}$ | \( 1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.17.ah_bn | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.19.ad_u | |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) | 2.23.a_q | |
| 29 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.29.ah_cc | |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) | 2.31.a_ci | |
| 37 | $D_{4}$ | \( 1 - T + 40 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.37.ab_bo | |
| 41 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.41.f_bh | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.43.d_aw | |
| 47 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.47.i_ca | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.53.ah_bu | |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) | 2.59.a_abm | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.e_ck | |
| 67 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.67.i_bi | |
| 71 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.71.ai_bq | |
| 73 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.73.b_dg | |
| 79 | $D_{4}$ | \( 1 - 2 T - 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.79.ac_adm | |
| 83 | $D_{4}$ | \( 1 - 19 T + 208 T^{2} - 19 p T^{3} + p^{2} T^{4} \) | 2.83.at_ia | |
| 89 | $D_{4}$ | \( 1 - T - 27 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.89.ab_abb | |
| 97 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.97.ak_de | |
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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
−19.6679227232, −19.1687844395, −18.5945819642, −18.0669742721, −17.7774940465, −16.9725635965, −16.4191061415, −15.9379138107, −15.2040931962, −14.4650318890, −13.7923572937, −13.3009950214, −12.6609675324, −12.0008038338, −11.3934994195, −10.2212626601, −10.0011394250, −9.24056418157, −8.19123900334, −7.59780337154, −6.45288273711, −5.40128604449, −4.79821105212, −3.37269439426, 3.37269439426, 4.79821105212, 5.40128604449, 6.45288273711, 7.59780337154, 8.19123900334, 9.24056418157, 10.0011394250, 10.2212626601, 11.3934994195, 12.0008038338, 12.6609675324, 13.3009950214, 13.7923572937, 14.4650318890, 15.2040931962, 15.9379138107, 16.4191061415, 16.9725635965, 17.7774940465, 18.0669742721, 18.5945819642, 19.1687844395, 19.6679227232