Dirichlet series
| L(s) = 1 | − 3-s − 4-s − 6·7-s + 2·11-s + 12-s − 9·13-s − 3·16-s + 5·17-s + 6·21-s − 2·25-s + 4·27-s + 6·28-s + 3·31-s − 2·33-s − 7·37-s + 9·39-s − 41-s − 7·43-s − 2·44-s + 3·47-s + 3·48-s + 24·49-s − 5·51-s + 9·52-s − 53-s + 4·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 2.26·7-s + 0.603·11-s + 0.288·12-s − 2.49·13-s − 3/4·16-s + 1.21·17-s + 1.30·21-s − 2/5·25-s + 0.769·27-s + 1.13·28-s + 0.538·31-s − 0.348·33-s − 1.15·37-s + 1.44·39-s − 0.156·41-s − 1.06·43-s − 0.301·44-s + 0.437·47-s + 0.433·48-s + 24/7·49-s − 0.700·51-s + 1.24·52-s − 0.137·53-s + 0.520·59-s − 1.28·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9093 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9093 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(9093\) = \(3 \cdot 7 \cdot 433\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.579777\) |
| Root analytic conductor: | \(0.872600\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 9093,\ (\ :1/2, 1/2),\ -1)\) |
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) | ||
| 433 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 26 T + p T^{2} ) \) | ||
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) | 2.2.a_b |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.5.a_c | |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.11.ac_s | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.j_bs | |
| 17 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.17.af_y | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) | 2.23.a_bi | |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.29.a_k | |
| 31 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.31.ad_ac | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.37.h_dc | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.b_ca | |
| 43 | $D_{4}$ | \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.43.h_bi | |
| 47 | $D_{4}$ | \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.47.ad_abi | |
| 53 | $D_{4}$ | \( 1 + T + 28 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.53.b_bc | |
| 59 | $D_{4}$ | \( 1 - 4 T - 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.59.ae_aba | |
| 61 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.61.k_dy | |
| 67 | $D_{4}$ | \( 1 - 5 T + 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.67.af_k | |
| 71 | $C_2^2$ | \( 1 + 3 T + 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.71.d_cw | |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) | 2.73.a_bi | |
| 79 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.79.g_ck | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.83.f_de | |
| 89 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.89.ag_bm | |
| 97 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.97.m_gs | |
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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
−16.9165055550, −16.5853444158, −16.1310035851, −15.5019832889, −15.1095313800, −14.2977318904, −14.0591354933, −13.3697255567, −12.8293392286, −12.3044064249, −12.1025805347, −11.6144464537, −10.4856899098, −10.0963131857, −9.67901665501, −9.34540893955, −8.65329623234, −7.66404707694, −6.87677662414, −6.76288521236, −5.82711585527, −5.18504432115, −4.39904209957, −3.41512886727, −2.64036149730, 0, 2.64036149730, 3.41512886727, 4.39904209957, 5.18504432115, 5.82711585527, 6.76288521236, 6.87677662414, 7.66404707694, 8.65329623234, 9.34540893955, 9.67901665501, 10.0963131857, 10.4856899098, 11.6144464537, 12.1025805347, 12.3044064249, 12.8293392286, 13.3697255567, 14.0591354933, 14.2977318904, 15.1095313800, 15.5019832889, 16.1310035851, 16.5853444158, 16.9165055550