Dirichlet series
| L(s) = 1 | − 3·4-s − 9·5-s + 9·7-s − 8-s − 9·11-s − 3·13-s + 3·16-s − 15·17-s + 27·20-s − 6·23-s + 33·25-s − 27·28-s − 15·29-s − 81·35-s + 6·37-s + 9·40-s + 6·41-s + 15·43-s + 27·44-s − 9·47-s + 24·49-s + 9·52-s − 6·53-s + 81·55-s − 9·56-s − 15·59-s + 3·61-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 4.02·5-s + 3.40·7-s − 0.353·8-s − 2.71·11-s − 0.832·13-s + 3/4·16-s − 3.63·17-s + 6.03·20-s − 1.25·23-s + 33/5·25-s − 5.10·28-s − 2.78·29-s − 13.6·35-s + 0.986·37-s + 1.42·40-s + 0.937·41-s + 2.28·43-s + 4.07·44-s − 1.31·47-s + 24/7·49-s + 1.24·52-s − 0.824·53-s + 10.9·55-s − 1.20·56-s − 1.95·59-s + 0.384·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.04902\times 10^{8}\) |
| Root analytic conductor: | \(5.09346\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 3^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|
| bad | 3 | \( 1 \) | |
| 19 | \( 1 \) | ||
| good | 2 | \( 1 + 3 T^{2} + T^{3} + 3 p T^{4} + 3 p T^{5} + p^{3} T^{6} + 3 p^{2} T^{7} + 3 p^{3} T^{8} + p^{3} T^{9} + 3 p^{4} T^{10} + p^{6} T^{12} \) | 6.2.a_d_b_g_g_i |
| 5 | \( 1 + 9 T + 48 T^{2} + 188 T^{3} + 609 T^{4} + 339 p T^{5} + 4076 T^{6} + 339 p^{2} T^{7} + 609 p^{2} T^{8} + 188 p^{3} T^{9} + 48 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.5.j_bw_hg_xl_cnf_gau | |
| 7 | \( 1 - 9 T + 57 T^{2} - 36 p T^{3} + 984 T^{4} - 3186 T^{5} + 9257 T^{6} - 3186 p T^{7} + 984 p^{2} T^{8} - 36 p^{4} T^{9} + 57 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.7.aj_cf_ajs_blw_aeso_nsb | |
| 11 | \( 1 + 9 T + 72 T^{2} + 324 T^{3} + 1413 T^{4} + 4311 T^{5} + 15868 T^{6} + 4311 p T^{7} + 1413 p^{2} T^{8} + 324 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.11.j_cu_mm_ccj_gjv_xmi | |
| 13 | \( 1 + 3 T + 66 T^{2} + 12 p T^{3} + 1929 T^{4} + 3657 T^{5} + 32297 T^{6} + 3657 p T^{7} + 1929 p^{2} T^{8} + 12 p^{4} T^{9} + 66 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.13.d_co_ga_cwf_fkr_bvuf | |
| 17 | \( 1 + 15 T + 138 T^{2} + 900 T^{3} + 4983 T^{4} + 24117 T^{5} + 105892 T^{6} + 24117 p T^{7} + 4983 p^{2} T^{8} + 900 p^{3} T^{9} + 138 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.17.p_fi_biq_hjr_bjrp_gaqu | |
| 23 | \( 1 + 6 T + 99 T^{2} + 395 T^{3} + 4065 T^{4} + 12417 T^{5} + 108158 T^{6} + 12417 p T^{7} + 4065 p^{2} T^{8} + 395 p^{3} T^{9} + 99 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.23.g_dv_pf_gaj_sjp_gdzy | |
| 29 | \( 1 + 15 T + 249 T^{2} + 2300 T^{3} + 21231 T^{4} + 136605 T^{5} + 861062 T^{6} + 136605 p T^{7} + 21231 p^{2} T^{8} + 2300 p^{3} T^{9} + 249 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.p_jp_dkm_bfkp_hucb_bwztu | |
| 31 | \( 1 + 150 T^{2} + 10 T^{3} + 10266 T^{4} + 750 T^{5} + 407343 T^{6} + 750 p T^{7} + 10266 p^{2} T^{8} + 10 p^{3} T^{9} + 150 p^{4} T^{10} + p^{6} T^{12} \) | 6.31.a_fu_k_pew_bcw_xepb | |
| 37 | \( 1 - 6 T + 138 T^{2} - 248 T^{3} + 5688 T^{4} + 14688 T^{5} + 146841 T^{6} + 14688 p T^{7} + 5688 p^{2} T^{8} - 248 p^{3} T^{9} + 138 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.ag_fi_ajo_iku_vsy_ijft | |
| 41 | \( 1 - 6 T + 159 T^{2} - 635 T^{3} + 11889 T^{4} - 35895 T^{5} + 577886 T^{6} - 35895 p T^{7} + 11889 p^{2} T^{8} - 635 p^{3} T^{9} + 159 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.ag_gd_ayl_rph_acbcp_bgwwk | |
| 43 | \( 1 - 15 T + 192 T^{2} - 1678 T^{3} + 15297 T^{4} - 108039 T^{5} + 788787 T^{6} - 108039 p T^{7} + 15297 p^{2} T^{8} - 1678 p^{3} T^{9} + 192 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.ap_hk_acmo_wqj_agdvj_bswvz | |
| 47 | \( 1 + 9 T + 6 p T^{2} + 1926 T^{3} + 32541 T^{4} + 171513 T^{5} + 2020408 T^{6} + 171513 p T^{7} + 32541 p^{2} T^{8} + 1926 p^{3} T^{9} + 6 p^{5} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.j_kw_cwc_bwdp_jtsr_ekyua | |
| 53 | \( 1 + 6 T + 219 T^{2} + 1323 T^{3} + 21777 T^{4} + 126951 T^{5} + 1374406 T^{6} + 126951 p T^{7} + 21777 p^{2} T^{8} + 1323 p^{3} T^{9} + 219 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.g_il_byx_bgfp_hfut_dafdu | |
| 59 | \( 1 + 15 T + 378 T^{2} + 3870 T^{3} + 55179 T^{4} + 421431 T^{5} + 4291516 T^{6} + 421431 p T^{7} + 55179 p^{2} T^{8} + 3870 p^{3} T^{9} + 378 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.p_oo_fsw_ddqh_xzkx_jkeki | |
| 61 | \( 1 - 3 T + 213 T^{2} - 584 T^{3} + 21186 T^{4} - 52218 T^{5} + 1453581 T^{6} - 52218 p T^{7} + 21186 p^{2} T^{8} - 584 p^{3} T^{9} + 213 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.ad_if_awm_bfiw_aczgk_desgz | |
| 67 | \( 1 - 18 T + 357 T^{2} - 4373 T^{3} + 54933 T^{4} - 510171 T^{5} + 4792818 T^{6} - 510171 p T^{7} + 54933 p^{2} T^{8} - 4373 p^{3} T^{9} + 357 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.as_nt_agmf_ddgv_abdarz_kmrze | |
| 71 | \( 1 + 15 T + 387 T^{2} + 4446 T^{3} + 64233 T^{4} + 581379 T^{5} + 5950702 T^{6} + 581379 p T^{7} + 64233 p^{2} T^{8} + 4446 p^{3} T^{9} + 387 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.p_ox_gpa_dran_bhcat_naove | |
| 73 | \( 1 + 6 T + 300 T^{2} + 2038 T^{3} + 43782 T^{4} + 286698 T^{5} + 3940299 T^{6} + 286698 p T^{7} + 43782 p^{2} T^{8} + 2038 p^{3} T^{9} + 300 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.73.g_lo_dak_cmty_qicw_iqevz | |
| 79 | \( 1 + 21 T + 420 T^{2} + 6402 T^{3} + 81501 T^{4} + 888861 T^{5} + 8552927 T^{6} + 888861 p T^{7} + 81501 p^{2} T^{8} + 6402 p^{3} T^{9} + 420 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.v_qe_jmg_eqor_byowz_ssqgt | |
| 83 | \( 1 - 3 T + 222 T^{2} - 1246 T^{3} + 34209 T^{4} - 157407 T^{5} + 3488216 T^{6} - 157407 p T^{7} + 34209 p^{2} T^{8} - 1246 p^{3} T^{9} + 222 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.ad_io_abvy_bypt_aiywd_hqmce | |
| 89 | \( 1 + 24 T + 531 T^{2} + 7099 T^{3} + 93819 T^{4} + 932253 T^{5} + 9844202 T^{6} + 932253 p T^{7} + 93819 p^{2} T^{8} + 7099 p^{3} T^{9} + 531 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.y_ul_knb_fiul_cbbbx_vocle | |
| 97 | \( 1 - 3 T + 306 T^{2} + 110 T^{3} + 38589 T^{4} + 171843 T^{5} + 3552945 T^{6} + 171843 p T^{7} + 38589 p^{2} T^{8} + 110 p^{3} T^{9} + 306 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.ad_lu_eg_cfcf_jufj_hudvt | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.94878760344798356976894159255, −4.58195199960031850052706250319, −4.52472215675713872973959730524, −4.50744205333782977285135270302, −4.42581430432478090244795157032, −4.21920254002293989154188027735, −4.12872075505458328183751923908, −3.85291124518724886613672324076, −3.78367622133806687854695966363, −3.71686977819220528259381968803, −3.69935624935296947930624663105, −3.65419832698165080038354238819, −3.03636985908432318079319446833, −2.86501108053582813416534652847, −2.86377844514094779232650155754, −2.58010749860453153228858284115, −2.55886918603598284297187516164, −2.37992142956395559594063837881, −2.13052696738092534303433129741, −1.95408295893388769779012911268, −1.90900820554912429454232933012, −1.44272690030107891134126914999, −1.35580244253470893880582628165, −1.24355077247316904456532233959, −0.888398102554146291835019554700, 0, 0, 0, 0, 0, 0, 0.888398102554146291835019554700, 1.24355077247316904456532233959, 1.35580244253470893880582628165, 1.44272690030107891134126914999, 1.90900820554912429454232933012, 1.95408295893388769779012911268, 2.13052696738092534303433129741, 2.37992142956395559594063837881, 2.55886918603598284297187516164, 2.58010749860453153228858284115, 2.86377844514094779232650155754, 2.86501108053582813416534652847, 3.03636985908432318079319446833, 3.65419832698165080038354238819, 3.69935624935296947930624663105, 3.71686977819220528259381968803, 3.78367622133806687854695966363, 3.85291124518724886613672324076, 4.12872075505458328183751923908, 4.21920254002293989154188027735, 4.42581430432478090244795157032, 4.50744205333782977285135270302, 4.52472215675713872973959730524, 4.58195199960031850052706250319, 4.94878760344798356976894159255
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.