Dirichlet series
| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·9-s + 2·10-s − 11-s − 13-s − 14-s − 3·16-s − 3·17-s + 2·18-s − 19-s − 2·20-s + 22-s − 23-s − 2·25-s + 26-s − 3·27-s + 28-s − 8·29-s + 2·31-s + 5·32-s + 3·34-s − 2·35-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.301·11-s − 0.277·13-s − 0.267·14-s − 3/4·16-s − 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.447·20-s + 0.213·22-s − 0.208·23-s − 2/5·25-s + 0.196·26-s − 0.577·27-s + 0.188·28-s − 1.48·29-s + 0.359·31-s + 0.883·32-s + 0.514·34-s − 0.338·35-s − 1/3·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 44178 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44178 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(44178\) = \(2 \cdot 3 \cdot 37 \cdot 199\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.81682\) |
| Root analytic conductor: | \(1.29550\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 44178,\ (\ :1/2, 1/2),\ -1)\) |
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_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) | ||
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) | ||
| 199 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) | ||
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_g |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.7.ab_i | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.b_k | |
| 13 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_i | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.19.b_g | |
| 23 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.23.b_i | |
| 29 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.29.i_bq | |
| 31 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.31.ac_k | |
| 41 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_bz | |
| 43 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.43.k_de | |
| 47 | $D_{4}$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.47.ag_af | |
| 53 | $D_{4}$ | \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.53.af_cc | |
| 59 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.59.m_dy | |
| 61 | $D_{4}$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.61.d_ba | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.67.ab_da | |
| 71 | $D_{4}$ | \( 1 + 9 T + 24 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.71.j_y | |
| 73 | $D_{4}$ | \( 1 - 5 T - 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.73.af_au | |
| 79 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.79.g_by | |
| 83 | $D_{4}$ | \( 1 - 5 T + 46 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.83.af_bu | |
| 89 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.89.ac_w | |
| 97 | $D_{4}$ | \( 1 - 2 T - 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.97.ac_ack | |
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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
−15.1833055115, −14.7990641822, −14.3106793130, −13.6540977418, −13.2659848486, −12.9367106892, −12.0552954840, −11.7221974625, −11.4118405304, −10.9291671297, −10.5700102709, −9.90330788995, −9.34392993641, −8.94919729310, −8.10142489859, −8.05022819352, −7.49493281569, −6.95526404047, −6.25151694800, −5.64717481853, −4.81607825355, −4.30577593587, −3.56160428141, −2.58343360055, −1.76748720559, 0, 1.76748720559, 2.58343360055, 3.56160428141, 4.30577593587, 4.81607825355, 5.64717481853, 6.25151694800, 6.95526404047, 7.49493281569, 8.05022819352, 8.10142489859, 8.94919729310, 9.34392993641, 9.90330788995, 10.5700102709, 10.9291671297, 11.4118405304, 11.7221974625, 12.0552954840, 12.9367106892, 13.2659848486, 13.6540977418, 14.3106793130, 14.7990641822, 15.1833055115