Dirichlet series
| L(s) = 1 | − 4·3-s − 4-s − 4·7-s + 7·9-s − 11-s + 4·12-s − 6·13-s + 16-s − 3·17-s − 2·19-s + 16·21-s − 2·23-s − 7·25-s − 4·27-s + 4·28-s − 5·29-s − 3·31-s + 4·33-s − 7·36-s − 2·37-s + 24·39-s − 4·41-s − 5·43-s + 44-s + 13·47-s − 4·48-s + 3·49-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1/2·4-s − 1.51·7-s + 7/3·9-s − 0.301·11-s + 1.15·12-s − 1.66·13-s + 1/4·16-s − 0.727·17-s − 0.458·19-s + 3.49·21-s − 0.417·23-s − 7/5·25-s − 0.769·27-s + 0.755·28-s − 0.928·29-s − 0.538·31-s + 0.696·33-s − 7/6·36-s − 0.328·37-s + 3.84·39-s − 0.624·41-s − 0.762·43-s + 0.150·44-s + 1.89·47-s − 0.577·48-s + 3/7·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 52252 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52252 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(52252\) = \(2^{2} \cdot 13063\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.33163\) |
| Root analytic conductor: | \(1.35102\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 52252,\ (\ :1/2, 1/2),\ 1)\) |
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 | $C_2$ | \( 1 + T^{2} \) | |
| 13063 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 14 T + p T^{2} ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.e_j |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) | 2.5.a_h | |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.7.e_n | |
| 11 | $D_{4}$ | \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.11.b_an | |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.13.g_be | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_4$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.19.c_ag | |
| 23 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.23.c_o | |
| 29 | $D_{4}$ | \( 1 + 5 T + 55 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.29.f_cd | |
| 31 | $D_{4}$ | \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.31.d_bb | |
| 37 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.37.c_v | |
| 41 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.41.e_q | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.43.f_by | |
| 47 | $D_{4}$ | \( 1 - 13 T + 97 T^{2} - 13 p T^{3} + p^{2} T^{4} \) | 2.47.an_dt | |
| 53 | $D_{4}$ | \( 1 + 11 T + 99 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.53.l_dv | |
| 59 | $D_{4}$ | \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.59.ad_x | |
| 61 | $D_{4}$ | \( 1 + 19 T + 207 T^{2} + 19 p T^{3} + p^{2} T^{4} \) | 2.61.t_hz | |
| 67 | $D_{4}$ | \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.67.ae_x | |
| 71 | $D_{4}$ | \( 1 - 15 T + 131 T^{2} - 15 p T^{3} + p^{2} T^{4} \) | 2.71.ap_fb | |
| 73 | $D_{4}$ | \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.73.ai_ba | |
| 79 | $C_4$ | \( 1 + 23 T + 279 T^{2} + 23 p T^{3} + p^{2} T^{4} \) | 2.79.x_kt | |
| 83 | $D_{4}$ | \( 1 - 5 T + 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.83.af_bi | |
| 89 | $D_{4}$ | \( 1 - 7 T + 65 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.89.ah_cn | |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.97.i_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
−15.4453964729, −14.8234797384, −14.1068125596, −13.6946172668, −13.1288798761, −12.6612462261, −12.4013096423, −12.0333419056, −11.5711871571, −10.9637320573, −10.7083818260, −10.0526896962, −9.69008553928, −9.39653165661, −8.63633728639, −7.79109384276, −7.26405205920, −6.67555666250, −6.24776573027, −5.76398394191, −5.31193216675, −4.75842594567, −4.11795549922, −3.24608517017, −2.13310204877, 0, 0, 2.13310204877, 3.24608517017, 4.11795549922, 4.75842594567, 5.31193216675, 5.76398394191, 6.24776573027, 6.67555666250, 7.26405205920, 7.79109384276, 8.63633728639, 9.39653165661, 9.69008553928, 10.0526896962, 10.7083818260, 10.9637320573, 11.5711871571, 12.0333419056, 12.4013096423, 12.6612462261, 13.1288798761, 13.6946172668, 14.1068125596, 14.8234797384, 15.4453964729