Dirichlet series
| L(s) = 1 | − 2·2-s − 3-s + 4-s + 2·6-s − 3·7-s − 12-s + 13-s + 6·14-s + 16-s − 19-s + 3·21-s + 25-s − 2·26-s + 27-s − 3·28-s + 6·29-s + 2·31-s + 2·32-s + 2·38-s − 39-s − 8·41-s − 6·42-s − 7·43-s + 2·47-s − 48-s + 2·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s + 0.816·6-s − 1.13·7-s − 0.288·12-s + 0.277·13-s + 1.60·14-s + 1/4·16-s − 0.229·19-s + 0.654·21-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.566·28-s + 1.11·29-s + 0.359·31-s + 0.353·32-s + 0.324·38-s − 0.160·39-s − 1.24·41-s − 0.925·42-s − 1.06·43-s + 0.291·47-s − 0.144·48-s + 2/7·49-s − 0.282·50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(74529\) = \(3^{2} \cdot 7^{2} \cdot 13^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(4.75203\) |
| Root analytic conductor: | \(1.47645\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 74529,\ (\ :1/2, 1/2),\ -1)\) |
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 | 3 | $C_2$ | \( 1 + T + T^{2} \) | |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) | ||
| 13 | $C_2$ | \( 1 - T + p T^{2} \) | ||
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.2.c_d |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) | 2.5.a_ab | |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.11.a_ad | |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) | 2.17.a_ba | |
| 19 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.19.b_ac | |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) | 2.23.a_bi | |
| 29 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.29.ag_br | |
| 31 | $D_{4}$ | \( 1 - 2 T + 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.31.ac_cd | |
| 37 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) | 2.37.a_f | |
| 41 | $D_{4}$ | \( 1 + 8 T + 39 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.41.i_bn | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.43.h_di | |
| 47 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.47.ac_bv | |
| 53 | $D_{4}$ | \( 1 + 4 T + 47 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.53.e_bv | |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) | 2.59.a_bm | |
| 61 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.61.ab_y | |
| 67 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.67.ae_bt | |
| 71 | $D_{4}$ | \( 1 - 18 T + 217 T^{2} - 18 p T^{3} + p^{2} T^{4} \) | 2.71.as_ij | |
| 73 | $D_{4}$ | \( 1 - 5 T - 5 p T^{3} + p^{2} T^{4} \) | 2.73.af_a | |
| 79 | $D_{4}$ | \( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.79.ak_dj | |
| 83 | $D_{4}$ | \( 1 - 16 T + 154 T^{2} - 16 p T^{3} + p^{2} T^{4} \) | 2.83.aq_fy | |
| 89 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.89.ai_de | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 18 T + p T^{2} ) \) | 2.97.r_gu | |
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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
−14.7661388796, −13.9148624283, −13.6665128426, −13.2880431957, −12.5205805258, −12.3509228428, −11.8743142098, −11.2901003066, −10.7583459612, −10.3863601106, −9.85094749081, −9.61356801060, −9.11915653060, −8.59947606631, −8.16243585389, −7.79946061678, −6.77735953729, −6.57887412719, −6.21919637860, −5.28476351789, −4.89234842970, −3.89292789711, −3.27781516460, −2.43764443580, −1.09009144240, 0, 1.09009144240, 2.43764443580, 3.27781516460, 3.89292789711, 4.89234842970, 5.28476351789, 6.21919637860, 6.57887412719, 6.77735953729, 7.79946061678, 8.16243585389, 8.59947606631, 9.11915653060, 9.61356801060, 9.85094749081, 10.3863601106, 10.7583459612, 11.2901003066, 11.8743142098, 12.3509228428, 12.5205805258, 13.2880431957, 13.6665128426, 13.9148624283, 14.7661388796