Dirichlet series
| L(s) = 1 | − 2·3-s + 4-s − 2·5-s − 7-s + 3·9-s − 4·11-s − 2·12-s − 4·13-s + 4·15-s + 16-s − 17-s − 2·20-s + 2·21-s − 23-s − 2·25-s − 10·27-s − 28-s − 2·29-s + 9·31-s + 8·33-s + 2·35-s + 3·36-s − 10·37-s + 8·39-s − 41-s − 4·44-s − 6·45-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 9-s − 1.20·11-s − 0.577·12-s − 1.10·13-s + 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.447·20-s + 0.436·21-s − 0.208·23-s − 2/5·25-s − 1.92·27-s − 0.188·28-s − 0.371·29-s + 1.61·31-s + 1.39·33-s + 0.338·35-s + 1/2·36-s − 1.64·37-s + 1.28·39-s − 0.156·41-s − 0.603·44-s − 0.894·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 15068 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15068 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(15068\) = \(2^{2} \cdot 3767\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.960748\) |
| Root analytic conductor: | \(0.990039\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 15068,\ (\ :1/2, 1/2),\ -1)\) |
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$ | \( ( 1 - T )( 1 + T ) \) | |
| 3767 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 54 T + p T^{2} ) \) | ||
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.3.c_b |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_g | |
| 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_ac | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.e_k | |
| 13 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.13.e_x | |
| 17 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.17.b_i | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.b_ak | |
| 29 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_c | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.31.aj_dc | |
| 37 | $D_{4}$ | \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.37.k_cx | |
| 41 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.41.b_i | |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | 2.43.a_ak | |
| 47 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.47.d_bc | |
| 53 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.53.e_de | |
| 59 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.59.ai_de | |
| 61 | $D_{4}$ | \( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.61.m_el | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.ae_dy | |
| 71 | $D_{4}$ | \( 1 - 3 T + 128 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.71.ad_ey | |
| 73 | $D_{4}$ | \( 1 - 5 T + 72 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.73.af_cu | |
| 79 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.79.ag_be | |
| 83 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) | 2.83.a_ct | |
| 89 | $D_{4}$ | \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.89.p_fq | |
| 97 | $D_{4}$ | \( 1 + T + 28 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.97.b_bc | |
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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
−16.1071806400, −15.8923317628, −15.5085597197, −15.1845656666, −14.6069438132, −13.6055206321, −13.4918137465, −12.6667464725, −12.1917046949, −12.0184375036, −11.4207479617, −10.8910531545, −10.5087881816, −9.80197748273, −9.58562567501, −8.37781772878, −7.94983751515, −7.32079407040, −6.93579407229, −6.17879618086, −5.50559026301, −4.94751535223, −4.19618338042, −3.26306418958, −2.16404677805, 0, 2.16404677805, 3.26306418958, 4.19618338042, 4.94751535223, 5.50559026301, 6.17879618086, 6.93579407229, 7.32079407040, 7.94983751515, 8.37781772878, 9.58562567501, 9.80197748273, 10.5087881816, 10.8910531545, 11.4207479617, 12.0184375036, 12.1917046949, 12.6667464725, 13.4918137465, 13.6055206321, 14.6069438132, 15.1845656666, 15.5085597197, 15.8923317628, 16.1071806400