Dirichlet series
| L(s) = 1 | − 3-s + 3·5-s + 4·7-s + 9-s + 11-s + 4·13-s − 3·15-s − 17-s − 7·19-s − 4·21-s + 5·23-s + 5·25-s − 4·27-s − 3·31-s − 33-s + 12·35-s − 3·37-s − 4·39-s − 4·41-s + 6·43-s + 3·45-s + 3·47-s + 9·49-s + 51-s + 5·53-s + 3·55-s + 7·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.774·15-s − 0.242·17-s − 1.60·19-s − 0.872·21-s + 1.04·23-s + 25-s − 0.769·27-s − 0.538·31-s − 0.174·33-s + 2.02·35-s − 0.493·37-s − 0.640·39-s − 0.624·41-s + 0.914·43-s + 0.447·45-s + 0.437·47-s + 9/7·49-s + 0.140·51-s + 0.686·53-s + 0.404·55-s + 0.927·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(100352\) = \(2^{11} \cdot 7^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.39853\) |
| Root analytic conductor: | \(1.59045\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 100352,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.034461667\) |
| \(L(\frac12)\) | \(\approx\) | \(2.034461667\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.b_a |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.5.ad_e | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.11.ab_q | |
| 13 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.13.ae_g | |
| 17 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.17.b_q | |
| 19 | $D_{4}$ | \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.19.h_bk | |
| 23 | $D_{4}$ | \( 1 - 5 T + 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.23.af_bi | |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.29.a_ac | |
| 31 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.31.d_ac | |
| 37 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.37.d_q | |
| 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$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.ag_bu | |
| 47 | $D_{4}$ | \( 1 - 3 T - 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.47.ad_aba | |
| 53 | $D_{4}$ | \( 1 - 5 T + 64 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.53.af_cm | |
| 59 | $D_{4}$ | \( 1 + 5 T + 76 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.59.f_cy | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.61.ad_ca | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.67.af_dg | |
| 71 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.71.am_dq | |
| 73 | $D_{4}$ | \( 1 + 5 T + 120 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.73.f_eq | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) | 2.79.av_je | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) | 2.83.be_os | |
| 89 | $D_{4}$ | \( 1 + 9 T + 160 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.89.j_ge | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) | 2.97.q_gc | |
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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
−13.9740632385, −13.7178720534, −13.1837955659, −12.7431669434, −12.4425181497, −11.7387700085, −11.1919029348, −10.9342404398, −10.7626910821, −10.0887643592, −9.60945182042, −8.98875301674, −8.60507792458, −8.31490858603, −7.51168587181, −6.92725447211, −6.47213886804, −5.92337212334, −5.44070361977, −5.04090050718, −4.28700977404, −3.83388412043, −2.59267474112, −1.85888778669, −1.28348966028, 1.28348966028, 1.85888778669, 2.59267474112, 3.83388412043, 4.28700977404, 5.04090050718, 5.44070361977, 5.92337212334, 6.47213886804, 6.92725447211, 7.51168587181, 8.31490858603, 8.60507792458, 8.98875301674, 9.60945182042, 10.0887643592, 10.7626910821, 10.9342404398, 11.1919029348, 11.7387700085, 12.4425181497, 12.7431669434, 13.1837955659, 13.7178720534, 13.9740632385