Dirichlet series
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 3·8-s − 9-s − 10-s + 2·11-s + 12-s − 14-s − 15-s + 16-s − 18-s − 20-s − 21-s + 2·22-s + 3·24-s + 25-s − 28-s − 2·29-s − 30-s − 9·31-s − 32-s + 2·33-s + 35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.612·24-s + 1/5·25-s − 0.188·28-s − 0.371·29-s − 0.182·30-s − 1.61·31-s − 0.176·32-s + 0.348·33-s + 0.169·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 14411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(14411\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.918857\) |
| Root analytic conductor: | \(0.979066\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 14411,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.764293984\) |
| \(L(\frac12)\) | \(\approx\) | \(1.764293984\) |
| \(L(\frac{3}{2})\) | 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 | 14411 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 180 T + p T^{2} ) \) | |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) | 2.2.ab_a |
| 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.3.ab_c | |
| 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.5.b_a | |
| 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_ac | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ac_w | |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) | 2.13.a_o | |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | 2.17.a_ak | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.a_as | |
| 29 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_bq | |
| 31 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.31.j_bu | |
| 37 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.37.g_by | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.e_cs | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.43.am_di | |
| 47 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.47.e_o | |
| 53 | $D_{4}$ | \( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.53.ag_c | |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) | 2.59.a_adu | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.61.l_ds | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.67.am_fy | |
| 71 | $D_{4}$ | \( 1 + 9 T + 102 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.71.j_dy | |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.73.ag_s | |
| 79 | $D_{4}$ | \( 1 + 7 T - 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.79.h_ac | |
| 83 | $D_{4}$ | \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.83.i_w | |
| 89 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.89.e_o | |
| 97 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.97.ai_di | |
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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
−15.9522281922, −15.6684595669, −15.1213945855, −14.5067879814, −14.2091492410, −13.8823792981, −13.1928072682, −12.8205313561, −12.3323022485, −11.6904754030, −11.2602731182, −10.7168988110, −10.2373008114, −9.38696372451, −8.95257087537, −8.43675734790, −7.58390884649, −7.23739813008, −6.63311807391, −5.80583930978, −5.15842458649, −4.26591629978, −3.75413618986, −2.95809608817, −1.86718229714, 1.86718229714, 2.95809608817, 3.75413618986, 4.26591629978, 5.15842458649, 5.80583930978, 6.63311807391, 7.23739813008, 7.58390884649, 8.43675734790, 8.95257087537, 9.38696372451, 10.2373008114, 10.7168988110, 11.2602731182, 11.6904754030, 12.3323022485, 12.8205313561, 13.1928072682, 13.8823792981, 14.2091492410, 14.5067879814, 15.1213945855, 15.6684595669, 15.9522281922