Dirichlet series
| L(s) = 1 | − 2-s − 3·3-s + 4-s − 4·5-s + 3·6-s − 4·7-s − 3·8-s + 2·9-s + 4·10-s − 4·11-s − 3·12-s + 13-s + 4·14-s + 12·15-s + 16-s − 2·18-s − 4·20-s + 12·21-s + 4·22-s − 3·23-s + 9·24-s + 6·25-s − 26-s + 6·27-s − 4·28-s − 4·29-s − 12·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.78·5-s + 1.22·6-s − 1.51·7-s − 1.06·8-s + 2/3·9-s + 1.26·10-s − 1.20·11-s − 0.866·12-s + 0.277·13-s + 1.06·14-s + 3.09·15-s + 1/4·16-s − 0.471·18-s − 0.894·20-s + 2.61·21-s + 0.852·22-s − 0.625·23-s + 1.83·24-s + 6/5·25-s − 0.196·26-s + 1.15·27-s − 0.755·28-s − 0.742·29-s − 2.19·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 18289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(18289\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.16612\) |
| Root analytic conductor: | \(1.03916\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 18289,\ (\ :1/2, 1/2),\ 1)\) |
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 | 18289 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 104 T + p T^{2} ) \) | |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.2.b_a |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.3.d_h | |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k | |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.7.e_i | |
| 11 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.11.e_u | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.13.ab_ae | |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) | 2.17.a_a | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 + 3 T - 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.23.d_an | |
| 29 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.29.e_i | |
| 31 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) | 2.31.a_g | |
| 37 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.37.d_y | |
| 41 | $D_{4}$ | \( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.41.ag_y | |
| 43 | $D_{4}$ | \( 1 + 3 T + 42 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.43.d_bq | |
| 47 | $D_{4}$ | \( 1 + 13 T + 115 T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.47.n_el | |
| 53 | $D_{4}$ | \( 1 + 5 T + 11 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.53.f_l | |
| 59 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.59.c_m | |
| 61 | $D_{4}$ | \( 1 + 5 T + 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.61.f_bq | |
| 67 | $D_{4}$ | \( 1 - T + 57 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.67.ab_cf | |
| 71 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.71.ag_cy | |
| 73 | $D_{4}$ | \( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.73.aj_ee | |
| 79 | $D_{4}$ | \( 1 + 11 T + 183 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.79.l_hb | |
| 83 | $D_{4}$ | \( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.83.p_fd | |
| 89 | $D_{4}$ | \( 1 + 17 T + 245 T^{2} + 17 p T^{3} + p^{2} T^{4} \) | 2.89.r_jl | |
| 97 | $D_{4}$ | \( 1 + 23 T + 299 T^{2} + 23 p T^{3} + p^{2} T^{4} \) | 2.97.x_ln | |
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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
−16.1363850496, −16.0843780816, −15.5825664724, −15.4072888502, −14.6978191774, −13.8968690828, −13.1518204961, −12.5642502024, −12.4009971926, −11.7919153114, −11.4106158756, −11.0535226854, −10.6962619005, −9.75265924769, −9.66736162690, −8.56895968526, −8.28605127068, −7.56968999605, −6.93865820532, −6.40194367410, −5.88484156799, −5.33606238765, −4.38834881036, −3.46325598917, −2.83926901858, 0, 0, 2.83926901858, 3.46325598917, 4.38834881036, 5.33606238765, 5.88484156799, 6.40194367410, 6.93865820532, 7.56968999605, 8.28605127068, 8.56895968526, 9.66736162690, 9.75265924769, 10.6962619005, 11.0535226854, 11.4106158756, 11.7919153114, 12.4009971926, 12.5642502024, 13.1518204961, 13.8968690828, 14.6978191774, 15.4072888502, 15.5825664724, 16.0843780816, 16.1363850496