Dirichlet series
| L(s) = 1 | − 3-s + 4-s − 5-s − 7-s − 9-s + 2·11-s − 12-s + 13-s + 15-s + 16-s − 7·17-s − 4·19-s − 20-s + 21-s − 2·23-s − 7·25-s − 28-s + 3·29-s − 2·33-s + 35-s − 36-s + 6·37-s − 39-s − 3·41-s − 43-s + 2·44-s + 45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.417·23-s − 7/5·25-s − 0.188·28-s + 0.557·29-s − 0.348·33-s + 0.169·35-s − 1/6·36-s + 0.986·37-s − 0.160·39-s − 0.468·41-s − 0.152·43-s + 0.301·44-s + 0.149·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 58492 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58492 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(58492\) = \(2^{2} \cdot 7 \cdot 2089\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.72950\) |
| Root analytic conductor: | \(1.38967\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 58492,\ (\ :1/2, 1/2),\ -1)\) |
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| 2089 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) | ||
| 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 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.ac_ac | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.13.ab_ae | |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.17.h_bg | |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.19.e_w | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.c_ac | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.29.ad_bo | |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) | 2.31.a_aby | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.37.ag_cg | |
| 41 | $D_{4}$ | \( 1 + 3 T + 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.41.d_bw | |
| 43 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.43.b_aw | |
| 47 | $D_{4}$ | \( 1 + 7 T + 102 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.47.h_dy | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.53.g_bi | |
| 59 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.59.c_bu | |
| 61 | $D_{4}$ | \( 1 - 5 T + 100 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.61.af_dw | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.67.c_eg | |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) | 2.71.aq_gs | |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.73.a_aby | |
| 79 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.79.g_da | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.83.g_dq | |
| 89 | $D_{4}$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.89.aj_ce | |
| 97 | $D_{4}$ | \( 1 + 6 T - 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.97.g_ao | |
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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
−15.0025188112, −14.3814338367, −13.8785815636, −13.4483518745, −12.8378570571, −12.6662461669, −11.8209756879, −11.6561166125, −11.2306957127, −10.9062757590, −10.3069051493, −9.68783197519, −9.33329941366, −8.60664154663, −8.18677705668, −7.74817166083, −6.83424622993, −6.55169832364, −6.18505072567, −5.56827217035, −4.70235947181, −4.15929815950, −3.55919743985, −2.59489737831, −1.77120504254, 0, 1.77120504254, 2.59489737831, 3.55919743985, 4.15929815950, 4.70235947181, 5.56827217035, 6.18505072567, 6.55169832364, 6.83424622993, 7.74817166083, 8.18677705668, 8.60664154663, 9.33329941366, 9.68783197519, 10.3069051493, 10.9062757590, 11.2306957127, 11.6561166125, 11.8209756879, 12.6662461669, 12.8378570571, 13.4483518745, 13.8785815636, 14.3814338367, 15.0025188112