Dirichlet series
| L(s) = 1 | − 2·2-s − 3·3-s − 3·5-s + 6·6-s + 7-s + 4·8-s + 4·9-s + 6·10-s − 5·11-s + 13-s − 2·14-s + 9·15-s − 4·16-s − 2·17-s − 8·18-s + 19-s − 3·21-s + 10·22-s + 23-s − 12·24-s − 2·25-s − 2·26-s − 6·27-s − 29-s − 18·30-s − 4·31-s + 15·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s − 1.34·5-s + 2.44·6-s + 0.377·7-s + 1.41·8-s + 4/3·9-s + 1.89·10-s − 1.50·11-s + 0.277·13-s − 0.534·14-s + 2.32·15-s − 16-s − 0.485·17-s − 1.88·18-s + 0.229·19-s − 0.654·21-s + 2.13·22-s + 0.208·23-s − 2.44·24-s − 2/5·25-s − 0.392·26-s − 1.15·27-s − 0.185·29-s − 3.28·30-s − 0.718·31-s + 2.61·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(971\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.0619118\) |
| Root analytic conductor: | \(0.498819\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 971,\ (\ :1/2, 1/2),\ -1)\) |
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 | 971 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 45 T + p T^{2} ) \) | |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.2.c_e |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.3.d_f | |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.5.d_l | |
| 7 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) | 2.7.ab_b | |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.11.f_o | |
| 13 | $D_{4}$ | \( 1 - T - 8 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.13.ab_ai | |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.17.c_c | |
| 19 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.19.ab_d | |
| 23 | $D_{4}$ | \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.23.ab_bj | |
| 29 | $D_{4}$ | \( 1 + T - 18 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.29.b_as | |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.31.e_bu | |
| 37 | $D_{4}$ | \( 1 - 2 T - 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.37.ac_aw | |
| 41 | $D_{4}$ | \( 1 - 2 T + 40 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.41.ac_bo | |
| 43 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.43.e_c | |
| 47 | $D_{4}$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.47.ad_bf | |
| 53 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.53.aj_cm | |
| 59 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.59.k_ew | |
| 61 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.61.af_bk | |
| 67 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.67.k_cs | |
| 71 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.71.k_eg | |
| 73 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) | 2.73.as_ic | |
| 79 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.79.c_bi | |
| 83 | $D_{4}$ | \( 1 + T - 62 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.83.b_ack | |
| 89 | $D_{4}$ | \( 1 + 18 T + 242 T^{2} + 18 p T^{3} + p^{2} T^{4} \) | 2.89.s_ji | |
| 97 | $D_{4}$ | \( 1 - 7 T + 13 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.97.ah_n | |
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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
−19.5818273357, −18.8718265009, −18.4401347463, −18.1216228392, −17.6059795431, −17.1803778131, −16.5239334225, −16.1328290709, −15.4661667363, −15.0236368909, −13.7046769929, −13.2419394583, −12.3793484813, −11.7460864033, −11.1338646355, −10.7891651399, −10.0766342030, −9.26486906177, −8.34145830164, −7.82595935972, −7.23029168673, −5.83743521332, −5.06166200050, −4.11958312596, 0, 4.11958312596, 5.06166200050, 5.83743521332, 7.23029168673, 7.82595935972, 8.34145830164, 9.26486906177, 10.0766342030, 10.7891651399, 11.1338646355, 11.7460864033, 12.3793484813, 13.2419394583, 13.7046769929, 15.0236368909, 15.4661667363, 16.1328290709, 16.5239334225, 17.1803778131, 17.6059795431, 18.1216228392, 18.4401347463, 18.8718265009, 19.5818273357