Dirichlet series
| L(s) = 1 | − 3-s − 5-s − 4·7-s − 9-s − 3·11-s + 15-s − 17-s + 19-s + 4·21-s − 23-s − 7·25-s + 4·29-s − 5·31-s + 3·33-s + 4·35-s − 7·37-s + 4·43-s + 45-s − 3·47-s + 9·49-s + 51-s − 3·53-s + 3·55-s − 57-s + 7·59-s + 11·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s − 1/3·9-s − 0.904·11-s + 0.258·15-s − 0.242·17-s + 0.229·19-s + 0.872·21-s − 0.208·23-s − 7/5·25-s + 0.742·29-s − 0.898·31-s + 0.522·33-s + 0.676·35-s − 1.15·37-s + 0.609·43-s + 0.149·45-s − 0.437·47-s + 9/7·49-s + 0.140·51-s − 0.412·53-s + 0.404·55-s − 0.132·57-s + 0.911·59-s + 1.40·61-s + 0.503·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(12544\) = \(2^{8} \cdot 7^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.799816\) |
| Root analytic conductor: | \(0.945687\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 12544,\ (\ :1/2, 1/2),\ -1)\) |
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 | \( 1 \) | ||
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) | ||
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_c |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.b_i | |
| 11 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.11.d_c | |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) | 2.13.a_g | |
| 17 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.17.b_m | |
| 19 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.19.ab_ag | |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.23.b_aw | |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.29.ae_be | |
| 31 | $D_{4}$ | \( 1 + 5 T + 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.31.f_bq | |
| 37 | $D_{4}$ | \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.37.h_bk | |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) | 2.41.a_be | |
| 43 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.43.ae_g | |
| 47 | $D_{4}$ | \( 1 + 3 T - 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.47.d_ag | |
| 53 | $D_{4}$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.53.d_au | |
| 59 | $D_{4}$ | \( 1 - 7 T + 114 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.59.ah_ek | |
| 61 | $D_{4}$ | \( 1 - 11 T + 56 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.61.al_ce | |
| 67 | $D_{4}$ | \( 1 - 17 T + 154 T^{2} - 17 p T^{3} + p^{2} T^{4} \) | 2.67.ar_fy | |
| 71 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.71.i_be | |
| 73 | $D_{4}$ | \( 1 - 3 T - 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.73.ad_abk | |
| 79 | $D_{4}$ | \( 1 + 17 T + 186 T^{2} + 17 p T^{3} + p^{2} T^{4} \) | 2.79.r_he | |
| 83 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.83.ae_bm | |
| 89 | $D_{4}$ | \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.89.ad_bs | |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) | 2.97.a_eg | |
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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
−16.2803151136, −16.0477416511, −15.7883927068, −15.3571362640, −14.5935605841, −14.0913255100, −13.4043430965, −13.1627701348, −12.4886366805, −12.2181084167, −11.4544148965, −11.1863736592, −10.4277162864, −9.91341150047, −9.60312574383, −8.73019020106, −8.25319553746, −7.45686411419, −6.96523418741, −6.25137441270, −5.68286246531, −5.12903627169, −4.01021899369, −3.39555020361, −2.41325366860, 0, 2.41325366860, 3.39555020361, 4.01021899369, 5.12903627169, 5.68286246531, 6.25137441270, 6.96523418741, 7.45686411419, 8.25319553746, 8.73019020106, 9.60312574383, 9.91341150047, 10.4277162864, 11.1863736592, 11.4544148965, 12.2181084167, 12.4886366805, 13.1627701348, 13.4043430965, 14.0913255100, 14.5935605841, 15.3571362640, 15.7883927068, 16.0477416511, 16.2803151136