Dirichlet series
| L(s) = 1 | − 2·2-s + 4-s + 2·5-s + 9-s − 4·10-s − 3·11-s + 5·13-s + 16-s + 6·17-s − 2·18-s + 3·19-s + 2·20-s + 6·22-s − 7·23-s − 2·25-s − 10·26-s − 12·29-s + 4·31-s + 2·32-s − 12·34-s + 36-s − 17·37-s − 6·38-s + 11·41-s − 14·43-s − 3·44-s + 2·45-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.894·5-s + 1/3·9-s − 1.26·10-s − 0.904·11-s + 1.38·13-s + 1/4·16-s + 1.45·17-s − 0.471·18-s + 0.688·19-s + 0.447·20-s + 1.27·22-s − 1.45·23-s − 2/5·25-s − 1.96·26-s − 2.22·29-s + 0.718·31-s + 0.353·32-s − 2.05·34-s + 1/6·36-s − 2.79·37-s − 0.973·38-s + 1.71·41-s − 2.13·43-s − 0.452·44-s + 0.298·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 177813 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177813 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(177813\) = \(3^{2} \cdot 23 \cdot 859\) |
| Sign: | $-1$ |
| Analytic conductor: | \(11.3375\) |
| Root analytic conductor: | \(1.83497\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 177813,\ (\ :1/2, 1/2),\ -1)\) |
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) | ||
| 859 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 12 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.2.c_d |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.5.ac_g | |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.a_ac | |
| 11 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.11.d_c | |
| 13 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.13.af_q | |
| 17 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.17.ag_w | |
| 19 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.19.ad_c | |
| 29 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.29.m_cw | |
| 31 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.31.ae_k | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.r_fo | |
| 41 | $D_{4}$ | \( 1 - 11 T + 96 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.41.al_ds | |
| 43 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.43.o_eo | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.aj_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.53.m_da | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.59.b_du | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.61.i_bm | |
| 67 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.67.m_eg | |
| 71 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.71.m_dq | |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.73.w_ji | |
| 79 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.79.c_ek | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) | 2.83.abb_ni | |
| 89 | $D_{4}$ | \( 1 + 3 T - 76 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.89.d_acy | |
| 97 | $D_{4}$ | \( 1 - T + 28 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.97.ab_bc | |
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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
−13.6425035927, −13.5314310016, −12.9532658606, −12.3663102778, −11.8916966237, −11.7058102996, −10.7626733161, −10.5671311414, −10.2024218426, −9.74343509385, −9.47840161981, −8.90574465578, −8.66623378320, −7.97184174423, −7.65248966594, −7.39670859702, −6.42571945023, −5.90959823849, −5.68940313400, −5.12213766533, −4.21367028939, −3.51092756523, −2.99896373659, −1.79853143732, −1.43558137450, 0, 1.43558137450, 1.79853143732, 2.99896373659, 3.51092756523, 4.21367028939, 5.12213766533, 5.68940313400, 5.90959823849, 6.42571945023, 7.39670859702, 7.65248966594, 7.97184174423, 8.66623378320, 8.90574465578, 9.47840161981, 9.74343509385, 10.2024218426, 10.5671311414, 10.7626733161, 11.7058102996, 11.8916966237, 12.3663102778, 12.9532658606, 13.5314310016, 13.6425035927