Dirichlet series
| L(s) = 1 | − 3·3-s − 3·7-s + 4·9-s − 3·11-s + 4·13-s − 4·16-s − 2·17-s − 12·19-s + 9·21-s − 5·23-s − 7·25-s − 6·27-s − 2·29-s − 4·31-s + 9·33-s − 3·37-s − 12·39-s + 4·41-s − 6·43-s + 6·47-s + 12·48-s + 2·49-s + 6·51-s + 5·53-s + 36·57-s + 4·59-s − 3·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 4/3·9-s − 0.904·11-s + 1.10·13-s − 16-s − 0.485·17-s − 2.75·19-s + 1.96·21-s − 1.04·23-s − 7/5·25-s − 1.15·27-s − 0.371·29-s − 0.718·31-s + 1.56·33-s − 0.493·37-s − 1.92·39-s + 0.624·41-s − 0.914·43-s + 0.875·47-s + 1.73·48-s + 2/7·49-s + 0.840·51-s + 0.686·53-s + 4.76·57-s + 0.520·59-s − 0.384·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 198517 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198517 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(198517\) = \(11 \cdot 18047\) |
| Sign: | $1$ |
| Analytic conductor: | \(12.6576\) |
| Root analytic conductor: | \(1.88620\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 198517,\ (\ :1/2, 1/2),\ 1)\) |
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | |
| 18047 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 170 T + p T^{2} ) \) | ||
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.2.a_a |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.3.d_f | |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) | 2.5.a_h | |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.7.d_h | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.13.ae_v | |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.17.c_w | |
| 19 | $D_{4}$ | \( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.19.m_cq | |
| 23 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.23.f_p | |
| 29 | $D_{4}$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_ab | |
| 31 | $D_{4}$ | \( 1 + 4 T - 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.31.e_am | |
| 37 | $D_{4}$ | \( 1 + 3 T - 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.37.d_az | |
| 41 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.41.ae_o | |
| 43 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.43.g_cl | |
| 47 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.47.ag_du | |
| 53 | $D_{4}$ | \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.53.af_cc | |
| 59 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.59.ae_q | |
| 61 | $D_{4}$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.61.d_dw | |
| 67 | $D_{4}$ | \( 1 + 8 T + 95 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.67.i_dr | |
| 71 | $D_{4}$ | \( 1 - 3 T - 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.71.ad_abj | |
| 73 | $D_{4}$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.73.e_d | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.79.e_du | |
| 83 | $D_{4}$ | \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.83.ad_bi | |
| 89 | $D_{4}$ | \( 1 + 5 T - 23 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.89.f_ax | |
| 97 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.97.d_k | |
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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
−13.5678306849, −13.2830984440, −12.9362963858, −12.6766638118, −12.0815262034, −11.6717915034, −11.2695196955, −10.9260990178, −10.4766218981, −10.2710808836, −9.71681719758, −8.96901721873, −8.81844685791, −8.11812020923, −7.57809901731, −6.92949842592, −6.34802988158, −6.26367752017, −5.78649121846, −5.35185273737, −4.52525123096, −4.06116935846, −3.63231188822, −2.41934404894, −1.91090273078, 0, 0, 1.91090273078, 2.41934404894, 3.63231188822, 4.06116935846, 4.52525123096, 5.35185273737, 5.78649121846, 6.26367752017, 6.34802988158, 6.92949842592, 7.57809901731, 8.11812020923, 8.81844685791, 8.96901721873, 9.71681719758, 10.2710808836, 10.4766218981, 10.9260990178, 11.2695196955, 11.6717915034, 12.0815262034, 12.6766638118, 12.9362963858, 13.2830984440, 13.5678306849