Dirichlet series
| L(s) = 1 | + 3-s − 3·7-s + 9-s − 11-s − 5·13-s − 17-s − 3·21-s − 2·23-s − 2·25-s + 27-s − 5·29-s + 6·31-s − 33-s − 4·37-s − 5·39-s + 2·41-s + 8·43-s + 3·47-s − 49-s − 51-s − 12·53-s + 6·59-s − 12·61-s − 3·63-s − 5·67-s − 2·69-s − 10·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.242·17-s − 0.654·21-s − 0.417·23-s − 2/5·25-s + 0.192·27-s − 0.928·29-s + 1.07·31-s − 0.174·33-s − 0.657·37-s − 0.800·39-s + 0.312·41-s + 1.21·43-s + 0.437·47-s − 1/7·49-s − 0.140·51-s − 1.64·53-s + 0.781·59-s − 1.53·61-s − 0.377·63-s − 0.610·67-s − 0.240·69-s − 1.18·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 100224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(100224\) = \(2^{7} \cdot 3^{3} \cdot 29\) |
| Sign: | $-1$ |
| Analytic conductor: | \(6.39036\) |
| Root analytic conductor: | \(1.58994\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 100224,\ (\ :1/2, 1/2),\ -1)\) |
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 | \( 1 \) | ||
| 3 | $C_1$ | \( 1 - T \) | ||
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) | ||
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.5.a_c |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.d_k | |
| 11 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.11.b_g | |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.13.f_bc | |
| 17 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.17.b_ac | |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.19.a_bm | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.c_ac | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.ag_ck | |
| 37 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.37.e_be | |
| 41 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.41.ac_k | |
| 43 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.43.ai_ck | |
| 47 | $D_{4}$ | \( 1 - 3 T + 78 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.47.ad_da | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.53.m_ec | |
| 59 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.59.ag_cs | |
| 61 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.61.m_da | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.67.f_be | |
| 71 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.71.k_eg | |
| 73 | $D_{4}$ | \( 1 + 8 T - 2 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.73.i_ac | |
| 79 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.79.m_fu | |
| 83 | $D_{4}$ | \( 1 - 2 T - 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.83.ac_acw | |
| 89 | $D_{4}$ | \( 1 + 7 T + 102 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.89.h_dy | |
| 97 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.97.g_fi | |
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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
−14.1247217080, −13.9865408107, −13.2512551590, −13.0687610386, −12.5908668543, −12.0979076049, −11.8880029734, −11.0694420090, −10.6891552289, −10.0671358818, −9.70188428070, −9.51960255800, −8.85326297822, −8.43103236377, −7.67564159418, −7.41187194234, −6.96701282111, −6.15809088122, −5.92715422594, −5.05559316447, −4.47132494466, −3.90532679908, −3.00798403796, −2.71828796111, −1.76343437938, 0, 1.76343437938, 2.71828796111, 3.00798403796, 3.90532679908, 4.47132494466, 5.05559316447, 5.92715422594, 6.15809088122, 6.96701282111, 7.41187194234, 7.67564159418, 8.43103236377, 8.85326297822, 9.51960255800, 9.70188428070, 10.0671358818, 10.6891552289, 11.0694420090, 11.8880029734, 12.0979076049, 12.5908668543, 13.0687610386, 13.2512551590, 13.9865408107, 14.1247217080