Dirichlet series
| L(s) = 1 | − 2-s − 3·3-s − 3·5-s + 3·6-s − 3·7-s + 8-s + 4·9-s + 3·10-s − 11-s − 2·13-s + 3·14-s + 9·15-s − 16-s − 2·17-s − 4·18-s − 12·19-s + 9·21-s + 22-s − 7·23-s − 3·24-s + 25-s + 2·26-s − 6·27-s + 9·29-s − 9·30-s + 5·31-s + 3·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1.34·5-s + 1.22·6-s − 1.13·7-s + 0.353·8-s + 4/3·9-s + 0.948·10-s − 0.301·11-s − 0.554·13-s + 0.801·14-s + 2.32·15-s − 1/4·16-s − 0.485·17-s − 0.942·18-s − 2.75·19-s + 1.96·21-s + 0.213·22-s − 1.45·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 1.15·27-s + 1.67·29-s − 1.64·30-s + 0.898·31-s + 0.522·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 33476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(33476\) = \(2^{2} \cdot 8369\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.13445\) |
| Root analytic conductor: | \(1.20870\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 33476,\ (\ :1/2, 1/2),\ 1)\) |
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 | $C_2$ | \( 1 + T + T^{2} \) | |
| 8369 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 66 T + p T^{2} ) \) | ||
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.3.d_f |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.5.d_i | |
| 7 | $D_{4}$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.7.d_f | |
| 11 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.11.b_e | |
| 13 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_u | |
| 17 | $D_{4}$ | \( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.17.c_l | |
| 19 | $D_{4}$ | \( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.19.m_cq | |
| 23 | $D_{4}$ | \( 1 + 7 T + 50 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.23.h_by | |
| 29 | $D_{4}$ | \( 1 - 9 T + 49 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.29.aj_bx | |
| 31 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.31.af_y | |
| 37 | $D_{4}$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.37.e_d | |
| 41 | $D_{4}$ | \( 1 + 9 T + 67 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.41.j_cp | |
| 43 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.43.g_bc | |
| 47 | $D_{4}$ | \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.47.m_ea | |
| 53 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_as | |
| 59 | $D_{4}$ | \( 1 - T - 57 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.59.ab_acf | |
| 61 | $D_{4}$ | \( 1 - 13 T + 160 T^{2} - 13 p T^{3} + p^{2} T^{4} \) | 2.61.an_ge | |
| 67 | $D_{4}$ | \( 1 + T - 25 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.67.b_az | |
| 71 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.71.ai_ce | |
| 73 | $D_{4}$ | \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.73.ag_bt | |
| 79 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.79.ae_w | |
| 83 | $C_2^2$ | \( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} \) | 2.83.q_gr | |
| 89 | $D_{4}$ | \( 1 + 8 T + 16 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.89.i_q | |
| 97 | $D_{4}$ | \( 1 + 9 T + 161 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.97.j_gf | |
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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
−15.7831469082, −15.3613566597, −15.1035877169, −14.3011888704, −13.6159200856, −13.1183823433, −12.5693621121, −12.2817390825, −11.8246952480, −11.4034068045, −10.9668578355, −10.3749804110, −10.0011897707, −9.74230258238, −8.61860290764, −8.27909951983, −8.00463621691, −6.87381541021, −6.67378051356, −6.28256904879, −5.47766824089, −4.66598773939, −4.25825983936, −3.49389777027, −2.18627426858, 0, 0, 2.18627426858, 3.49389777027, 4.25825983936, 4.66598773939, 5.47766824089, 6.28256904879, 6.67378051356, 6.87381541021, 8.00463621691, 8.27909951983, 8.61860290764, 9.74230258238, 10.0011897707, 10.3749804110, 10.9668578355, 11.4034068045, 11.8246952480, 12.2817390825, 12.5693621121, 13.1183823433, 13.6159200856, 14.3011888704, 15.1035877169, 15.3613566597, 15.7831469082