Dirichlet series
| L(s) = 1 | + 4-s − 5·7-s − 9-s − 2·11-s − 2·13-s + 16-s − 3·17-s − 23-s − 7·25-s − 5·28-s − 8·29-s − 2·31-s − 36-s + 5·41-s + 4·43-s − 2·44-s + 3·47-s + 7·49-s − 2·52-s + 8·53-s + 4·59-s + 5·63-s + 64-s + 6·67-s − 3·68-s + 6·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.88·7-s − 1/3·9-s − 0.603·11-s − 0.554·13-s + 1/4·16-s − 0.727·17-s − 0.208·23-s − 7/5·25-s − 0.944·28-s − 1.48·29-s − 0.359·31-s − 1/6·36-s + 0.780·41-s + 0.609·43-s − 0.301·44-s + 0.437·47-s + 49-s − 0.277·52-s + 1.09·53-s + 0.520·59-s + 0.629·63-s + 1/8·64-s + 0.733·67-s − 0.363·68-s + 0.712·71-s − 0.234·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 35676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(35676\) = \(2^{2} \cdot 3^{2} \cdot 991\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.27473\) |
| Root analytic conductor: | \(1.22809\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 35676,\ (\ :1/2, 1/2),\ -1)\) |
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 3 | $C_2$ | \( 1 + T^{2} \) | ||
| 991 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 44 T + p T^{2} ) \) | ||
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) | 2.5.a_h |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.f_s | |
| 11 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.11.c_m | |
| 13 | $D_{4}$ | \( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_r | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.19.a_bl | |
| 23 | $D_{4}$ | \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.23.b_w | |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.29.i_cc | |
| 31 | $D_{4}$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.31.c_abh | |
| 37 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) | 2.37.a_u | |
| 41 | $D_{4}$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.41.af_ae | |
| 43 | $D_{4}$ | \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.43.ae_ag | |
| 47 | $D_{4}$ | \( 1 - 3 T - 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.47.ad_aba | |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.53.ai_cs | |
| 59 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.59.ae_z | |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) | 2.61.a_ek | |
| 67 | $D_{4}$ | \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.67.ag_l | |
| 71 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.71.ag_bj | |
| 73 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.73.c_fa | |
| 79 | $D_{4}$ | \( 1 + 3 T + 134 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.79.d_fe | |
| 83 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.83.e_bo | |
| 89 | $D_{4}$ | \( 1 - T + 144 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.89.ab_fo | |
| 97 | $D_{4}$ | \( 1 + 11 T + 198 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.97.l_hq | |
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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
−15.4279312609, −14.9840943234, −14.4875989485, −13.8108813922, −13.4337361656, −12.9444125971, −12.6671353324, −12.1725174174, −11.5648692248, −11.1110779020, −10.6343005993, −9.92239385669, −9.72436875293, −9.21954666239, −8.63633476756, −7.83085281161, −7.41141323679, −6.84975321071, −6.29936673934, −5.75466772703, −5.30802305995, −4.12222060581, −3.63679188343, −2.75852869818, −2.19038689356, 0, 2.19038689356, 2.75852869818, 3.63679188343, 4.12222060581, 5.30802305995, 5.75466772703, 6.29936673934, 6.84975321071, 7.41141323679, 7.83085281161, 8.63633476756, 9.21954666239, 9.72436875293, 9.92239385669, 10.6343005993, 11.1110779020, 11.5648692248, 12.1725174174, 12.6671353324, 12.9444125971, 13.4337361656, 13.8108813922, 14.4875989485, 14.9840943234, 15.4279312609