Dirichlet series
| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 3·8-s − 4·9-s − 10-s + 4·11-s − 7·13-s − 14-s + 16-s − 3·17-s + 4·18-s − 19-s + 20-s − 4·22-s − 8·23-s − 3·25-s + 7·26-s + 28-s + 29-s + 3·31-s + 32-s + 3·34-s + 35-s − 4·36-s − 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 1.06·8-s − 4/3·9-s − 0.316·10-s + 1.20·11-s − 1.94·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.942·18-s − 0.229·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s − 3/5·25-s + 1.37·26-s + 0.188·28-s + 0.185·29-s + 0.538·31-s + 0.176·32-s + 0.514·34-s + 0.169·35-s − 2/3·36-s − 0.328·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 40447 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40447 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(40447\) = \(11 \cdot 3677\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.57893\) |
| Root analytic conductor: | \(1.26724\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 40447,\ (\ :1/2, 1/2),\ -1)\) |
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) | |
| 3677 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 30 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.2.b_a |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) | 2.3.a_e | |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.ab_e | |
| 7 | $D_{4}$ | \( 1 - T - p T^{2} - p T^{3} + p^{2} T^{4} \) | 2.7.ab_ah | |
| 13 | $D_{4}$ | \( 1 + 7 T + 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.13.h_bc | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.19.b_a | |
| 23 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.23.i_by | |
| 29 | $D_{4}$ | \( 1 - T - 7 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.29.ab_ah | |
| 31 | $D_{4}$ | \( 1 - 3 T + T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.31.ad_b | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.37.c_co | |
| 41 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.41.ac_bq | |
| 43 | $D_{4}$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.43.ab_ak | |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.47.e_bu | |
| 53 | $D_{4}$ | \( 1 + 13 T + 90 T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.53.n_dm | |
| 59 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.59.e_g | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.61.q_fu | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.67.e_dy | |
| 71 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.71.ah_cs | |
| 73 | $D_{4}$ | \( 1 - 9 T + 67 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.73.aj_cp | |
| 79 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.79.ac_m | |
| 83 | $D_{4}$ | \( 1 - 7 T + 77 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.83.ah_cz | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.89.ac_fy | |
| 97 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.97.ad_dt | |
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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
−15.1235631721, −14.7727385919, −14.2636102431, −13.9436258036, −13.6812305840, −12.6005247485, −12.2877890260, −11.8294582481, −11.6071901620, −11.0468548548, −10.4308511760, −9.82418918463, −9.45292314597, −9.03478146166, −8.61957338595, −7.90585488368, −7.57873287204, −6.67461718396, −6.22800215722, −5.91774367607, −4.97745610926, −4.42887283139, −3.39222450316, −2.49436704998, −2.00054620936, 0, 2.00054620936, 2.49436704998, 3.39222450316, 4.42887283139, 4.97745610926, 5.91774367607, 6.22800215722, 6.67461718396, 7.57873287204, 7.90585488368, 8.61957338595, 9.03478146166, 9.45292314597, 9.82418918463, 10.4308511760, 11.0468548548, 11.6071901620, 11.8294582481, 12.2877890260, 12.6005247485, 13.6812305840, 13.9436258036, 14.2636102431, 14.7727385919, 15.1235631721