Dirichlet series
| L(s) = 1 | − 2-s − 3-s − 4-s − 3·5-s + 6-s + 8-s − 9-s + 3·10-s + 3·11-s + 12-s + 13-s + 3·15-s + 3·16-s − 3·17-s + 18-s + 3·20-s − 3·22-s + 8·23-s − 24-s + 2·25-s − 26-s − 29-s − 3·30-s + 31-s − 3·32-s − 3·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s − 1/3·9-s + 0.948·10-s + 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.774·15-s + 3/4·16-s − 0.727·17-s + 0.235·18-s + 0.670·20-s − 0.639·22-s + 1.66·23-s − 0.204·24-s + 2/5·25-s − 0.196·26-s − 0.185·29-s − 0.547·30-s + 0.179·31-s − 0.530·32-s − 0.522·33-s + 0.514·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 44830 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44830 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(44830\) = \(2 \cdot 5 \cdot 4483\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.85839\) |
| Root analytic conductor: | \(1.30026\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 44830,\ (\ :1/2, 1/2),\ -1)\) |
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$ | \( ( 1 + T )( 1 + p T^{2} ) \) | |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| 4483 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 20 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_c |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) | 2.7.a_g | |
| 11 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.11.ad_t | |
| 13 | $D_{4}$ | \( 1 - T + 17 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.13.ab_r | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) | 2.19.a_be | |
| 23 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.23.ai_bw | |
| 29 | $D_{4}$ | \( 1 + T - p T^{2} + p T^{3} + p^{2} T^{4} \) | 2.29.b_abd | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.31.ab_bq | |
| 37 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.37.ac_ao | |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.41.ac_c | |
| 43 | $D_{4}$ | \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.43.ac_bw | |
| 47 | $D_{4}$ | \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.47.af_cf | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) | 2.53.n_ey | |
| 59 | $D_{4}$ | \( 1 + 15 T + 145 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.59.p_fp | |
| 61 | $D_{4}$ | \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.61.ad_bs | |
| 67 | $D_{4}$ | \( 1 + 7 T + 71 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.67.h_ct | |
| 71 | $D_{4}$ | \( 1 - 17 T + 185 T^{2} - 17 p T^{3} + p^{2} T^{4} \) | 2.71.ar_hd | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.73.j_bi | |
| 79 | $D_{4}$ | \( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.79.aj_cy | |
| 83 | $D_{4}$ | \( 1 - 6 T + 12 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.83.ag_m | |
| 89 | $D_{4}$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.89.h_m | |
| 97 | $D_{4}$ | \( 1 + 7 T + 58 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.97.h_cg | |
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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
−15.2702555537, −14.7119156753, −14.2686479673, −13.7322742102, −13.1990375095, −12.6263365516, −12.2901480574, −11.7623269382, −11.3103893317, −10.9388034266, −10.6707325583, −9.71211142229, −9.26418902818, −9.02973763763, −8.39289277072, −7.90265501128, −7.51991324791, −6.71727118029, −6.32109732610, −5.54396767258, −4.84229261813, −4.26336745735, −3.65828141217, −2.91180792538, −1.25664965429, 0, 1.25664965429, 2.91180792538, 3.65828141217, 4.26336745735, 4.84229261813, 5.54396767258, 6.32109732610, 6.71727118029, 7.51991324791, 7.90265501128, 8.39289277072, 9.02973763763, 9.26418902818, 9.71211142229, 10.6707325583, 10.9388034266, 11.3103893317, 11.7623269382, 12.2901480574, 12.6263365516, 13.1990375095, 13.7322742102, 14.2686479673, 14.7119156753, 15.2702555537