Dirichlet series
| L(s) = 1 | + 2·3-s − 2·4-s − 4·5-s + 2·7-s − 3·9-s − 2·11-s − 4·12-s − 4·13-s − 8·15-s + 4·16-s + 8·20-s + 4·21-s + 12·23-s + 6·25-s − 14·27-s − 4·28-s − 12·29-s − 8·31-s − 4·33-s − 8·35-s + 6·36-s + 2·37-s − 8·39-s − 18·41-s + 4·43-s + 4·44-s + 12·45-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 1.78·5-s + 0.755·7-s − 9-s − 0.603·11-s − 1.15·12-s − 1.10·13-s − 2.06·15-s + 16-s + 1.78·20-s + 0.872·21-s + 2.50·23-s + 6/5·25-s − 2.69·27-s − 0.755·28-s − 2.22·29-s − 1.43·31-s − 0.696·33-s − 1.35·35-s + 36-s + 0.328·37-s − 1.28·39-s − 2.81·41-s + 0.609·43-s + 0.603·44-s + 1.78·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(21904\) = \(2^{4} \cdot 37^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.39661\) |
| Root analytic conductor: | \(1.08709\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 21904,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + p T^{2} \) | |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) | ||
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) | 2.3.ac_h |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.7.ac_l | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.11.c_h | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.13.e_ba | |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.a_ac | |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.19.a_bi | |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.23.am_de | |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.29.m_dq | |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.31.i_da | |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) | 2.41.s_gh | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.ae_cc | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.47.ak_el | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.53.ag_db | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.59.aq_gk | |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.61.a_cg | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.ai_di | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.71.s_hf | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.73.ag_dn | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.79.q_ik | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) | 2.83.ak_gt | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.89.ai_hi | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.97.ai_hm | |
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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
−15.4804132508, −15.1978176633, −14.8059546844, −14.6280958917, −14.2687617053, −13.4643580848, −13.0450433708, −12.8187428343, −11.7961223953, −11.6538652075, −11.1833016517, −10.6206613214, −9.81996681750, −9.03234585169, −8.87881945327, −8.45778768172, −7.73434658622, −7.59911177067, −7.07951791765, −5.44973416215, −5.32652803781, −4.53527407985, −3.50910294340, −3.46400857722, −2.32683032801, 0, 2.32683032801, 3.46400857722, 3.50910294340, 4.53527407985, 5.32652803781, 5.44973416215, 7.07951791765, 7.59911177067, 7.73434658622, 8.45778768172, 8.87881945327, 9.03234585169, 9.81996681750, 10.6206613214, 11.1833016517, 11.6538652075, 11.7961223953, 12.8187428343, 13.0450433708, 13.4643580848, 14.2687617053, 14.6280958917, 14.8059546844, 15.1978176633, 15.4804132508