Dirichlet series
| L(s) = 1 | − 2-s − 3·3-s + 4-s − 5·5-s + 3·6-s − 7-s − 8-s + 4·9-s + 5·10-s − 3·12-s − 13-s + 14-s + 15·15-s + 16-s − 3·17-s − 4·18-s + 19-s − 5·20-s + 3·21-s + 2·23-s + 3·24-s + 12·25-s + 26-s − 6·27-s − 28-s − 15·30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 2.23·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 4/3·9-s + 1.58·10-s − 0.866·12-s − 0.277·13-s + 0.267·14-s + 3.87·15-s + 1/4·16-s − 0.727·17-s − 0.942·18-s + 0.229·19-s − 1.11·20-s + 0.654·21-s + 0.417·23-s + 0.612·24-s + 12/5·25-s + 0.196·26-s − 1.15·27-s − 0.188·28-s − 2.73·30-s − 0.359·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1952\) = \(2^{5} \cdot 61\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.124461\) |
| Root analytic conductor: | \(0.593961\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 1952,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 + T \) | |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) | ||
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.3.d_f |
| 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.5.f_n | |
| 7 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.7.b_a | |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.11.a_ac | |
| 13 | $D_{4}$ | \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_ah | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.19.ab_ak | |
| 23 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.23.ac_bi | |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) | 2.29.a_bc | |
| 31 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.31.c_c | |
| 37 | $D_{4}$ | \( 1 - 2 T - 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.37.ac_aq | |
| 41 | $D_{4}$ | \( 1 + 3 T - 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.41.d_ar | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.43.h_da | |
| 47 | $D_{4}$ | \( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.47.j_ec | |
| 53 | $D_{4}$ | \( 1 + 2 T + 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_bc | |
| 59 | $D_{4}$ | \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.59.ad_bi | |
| 67 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.67.ai_dy | |
| 71 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.71.ai_cs | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.73.h_cq | |
| 79 | $D_{4}$ | \( 1 + 16 T + 216 T^{2} + 16 p T^{3} + p^{2} T^{4} \) | 2.79.q_ii | |
| 83 | $D_{4}$ | \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.83.ad_bi | |
| 89 | $D_{4}$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.89.aj_cs | |
| 97 | $D_{4}$ | \( 1 - 7 T + 177 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.97.ah_gv | |
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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
−18.9369246633, −18.7579224684, −18.0634576052, −17.4490899324, −17.0398735652, −16.3815378081, −16.1730018779, −15.5639758026, −15.1960906504, −14.5705617732, −13.3072855395, −12.7217110610, −11.9917197633, −11.7158114248, −11.2748621525, −10.8818273211, −10.0945137999, −9.24368020508, −8.32219958616, −7.74114480575, −7.00109244261, −6.46627799353, −5.37865241307, −4.49147914724, −3.46784898938, 0, 3.46784898938, 4.49147914724, 5.37865241307, 6.46627799353, 7.00109244261, 7.74114480575, 8.32219958616, 9.24368020508, 10.0945137999, 10.8818273211, 11.2748621525, 11.7158114248, 11.9917197633, 12.7217110610, 13.3072855395, 14.5705617732, 15.1960906504, 15.5639758026, 16.1730018779, 16.3815378081, 17.0398735652, 17.4490899324, 18.0634576052, 18.7579224684, 18.9369246633