Properties

Label 6-650e3-1.1-c5e3-0-1
Degree 66
Conductor 274625000274625000
Sign 11
Analytic cond. 1.13297×1061.13297\times 10^{6}
Root an. cond. 10.210210.2102
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·2-s + 22·3-s + 96·4-s + 264·6-s + 234·7-s + 640·8-s − 45·9-s + 48·11-s + 2.11e3·12-s − 507·13-s + 2.80e3·14-s + 3.84e3·16-s − 1.50e3·17-s − 540·18-s − 360·19-s + 5.14e3·21-s + 576·22-s + 2.37e3·23-s + 1.40e4·24-s − 6.08e3·26-s − 2.93e3·27-s + 2.24e4·28-s − 3.07e3·29-s − 5.38e3·31-s + 2.15e4·32-s + 1.05e3·33-s − 1.80e4·34-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.41·3-s + 3·4-s + 2.99·6-s + 1.80·7-s + 3.53·8-s − 0.185·9-s + 0.119·11-s + 4.23·12-s − 0.832·13-s + 3.82·14-s + 15/4·16-s − 1.26·17-s − 0.392·18-s − 0.228·19-s + 2.54·21-s + 0.253·22-s + 0.934·23-s + 4.98·24-s − 1.76·26-s − 0.774·27-s + 5.41·28-s − 0.679·29-s − 1.00·31-s + 3.71·32-s + 0.168·33-s − 2.68·34-s + ⋯

Functional equation

Λ(s)=((2356133)s/2ΓC(s)3L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((2356133)s/2ΓC(s+5/2)3L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 23561332^{3} \cdot 5^{6} \cdot 13^{3}
Sign: 11
Analytic conductor: 1.13297×1061.13297\times 10^{6}
Root analytic conductor: 10.210210.2102
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 2356133, ( :5/2,5/2,5/2), 1)(6,\ 2^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) \approx 55.4709391355.47093913
L(12)L(\frac12) \approx 55.4709391355.47093913
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1p2T)3 ( 1 - p^{2} T )^{3}
5 1 1
13C1C_1 (1+p2T)3 ( 1 + p^{2} T )^{3}
good3S4×C2S_4\times C_2 122T+529T23232pT3+529p5T422p10T5+p15T6 1 - 22 T + 529 T^{2} - 3232 p T^{3} + 529 p^{5} T^{4} - 22 p^{10} T^{5} + p^{15} T^{6}
7S4×C2S_4\times C_2 1234T+48705T27824532T3+48705p5T4234p10T5+p15T6 1 - 234 T + 48705 T^{2} - 7824532 T^{3} + 48705 p^{5} T^{4} - 234 p^{10} T^{5} + p^{15} T^{6}
11S4×C2S_4\times C_2 148T+76377T2+58976052T3+76377p5T448p10T5+p15T6 1 - 48 T + 76377 T^{2} + 58976052 T^{3} + 76377 p^{5} T^{4} - 48 p^{10} T^{5} + p^{15} T^{6}
17S4×C2S_4\times C_2 1+1506T+3273807T2+2997817500T3+3273807p5T4+1506p10T5+p15T6 1 + 1506 T + 3273807 T^{2} + 2997817500 T^{3} + 3273807 p^{5} T^{4} + 1506 p^{10} T^{5} + p^{15} T^{6}
19S4×C2S_4\times C_2 1+360T+161955pT21434259964T3+161955p6T4+360p10T5+p15T6 1 + 360 T + 161955 p T^{2} - 1434259964 T^{3} + 161955 p^{6} T^{4} + 360 p^{10} T^{5} + p^{15} T^{6}
23S4×C2S_4\times C_2 12370T+19159293T230182176920T3+19159293p5T42370p10T5+p15T6 1 - 2370 T + 19159293 T^{2} - 30182176920 T^{3} + 19159293 p^{5} T^{4} - 2370 p^{10} T^{5} + p^{15} T^{6}
29S4×C2S_4\times C_2 1+3078T+41304531T2+146324754036T3+41304531p5T4+3078p10T5+p15T6 1 + 3078 T + 41304531 T^{2} + 146324754036 T^{3} + 41304531 p^{5} T^{4} + 3078 p^{10} T^{5} + p^{15} T^{6}
31S4×C2S_4\times C_2 1+5388T+26264277T2+161901418108T3+26264277p5T4+5388p10T5+p15T6 1 + 5388 T + 26264277 T^{2} + 161901418108 T^{3} + 26264277 p^{5} T^{4} + 5388 p^{10} T^{5} + p^{15} T^{6}
37S4×C2S_4\times C_2 125362T+395488467T23869148349420T3+395488467p5T425362p10T5+p15T6 1 - 25362 T + 395488467 T^{2} - 3869148349420 T^{3} + 395488467 p^{5} T^{4} - 25362 p^{10} T^{5} + p^{15} T^{6}
41S4×C2S_4\times C_2 1+15906T72143817T23142322038788T372143817p5T4+15906p10T5+p15T6 1 + 15906 T - 72143817 T^{2} - 3142322038788 T^{3} - 72143817 p^{5} T^{4} + 15906 p^{10} T^{5} + p^{15} T^{6}
43S4×C2S_4\times C_2 139306T+941655657T213605136511296T3+941655657p5T439306p10T5+p15T6 1 - 39306 T + 941655657 T^{2} - 13605136511296 T^{3} + 941655657 p^{5} T^{4} - 39306 p^{10} T^{5} + p^{15} T^{6}
47S4×C2S_4\times C_2 117778T+430606137T23892325349540T3+430606137p5T417778p10T5+p15T6 1 - 17778 T + 430606137 T^{2} - 3892325349540 T^{3} + 430606137 p^{5} T^{4} - 17778 p^{10} T^{5} + p^{15} T^{6}
53S4×C2S_4\times C_2 1+9246T139832133T2+2191133695476T3139832133p5T4+9246p10T5+p15T6 1 + 9246 T - 139832133 T^{2} + 2191133695476 T^{3} - 139832133 p^{5} T^{4} + 9246 p^{10} T^{5} + p^{15} T^{6}
59S4×C2S_4\times C_2 1+77760T+3729319041T2+118237047820380T3+3729319041p5T4+77760p10T5+p15T6 1 + 77760 T + 3729319041 T^{2} + 118237047820380 T^{3} + 3729319041 p^{5} T^{4} + 77760 p^{10} T^{5} + p^{15} T^{6}
61S4×C2S_4\times C_2 117982T+1546877139T216008491590916T3+1546877139p5T417982p10T5+p15T6 1 - 17982 T + 1546877139 T^{2} - 16008491590916 T^{3} + 1546877139 p^{5} T^{4} - 17982 p^{10} T^{5} + p^{15} T^{6}
67S4×C2S_4\times C_2 12922T+1163644365T257141979467316T3+1163644365p5T42922p10T5+p15T6 1 - 2922 T + 1163644365 T^{2} - 57141979467316 T^{3} + 1163644365 p^{5} T^{4} - 2922 p^{10} T^{5} + p^{15} T^{6}
71S4×C2S_4\times C_2 1+4944T+3669025341T2+114486868308T3+3669025341p5T4+4944p10T5+p15T6 1 + 4944 T + 3669025341 T^{2} + 114486868308 T^{3} + 3669025341 p^{5} T^{4} + 4944 p^{10} T^{5} + p^{15} T^{6}
73S4×C2S_4\times C_2 1+43278T+5189491839T2+177703264664708T3+5189491839p5T4+43278p10T5+p15T6 1 + 43278 T + 5189491839 T^{2} + 177703264664708 T^{3} + 5189491839 p^{5} T^{4} + 43278 p^{10} T^{5} + p^{15} T^{6}
79S4×C2S_4\times C_2 142120T+5223395229T2282172460852048T3+5223395229p5T442120p10T5+p15T6 1 - 42120 T + 5223395229 T^{2} - 282172460852048 T^{3} + 5223395229 p^{5} T^{4} - 42120 p^{10} T^{5} + p^{15} T^{6}
83S4×C2S_4\times C_2 1+58098T+10357081509T2+373431668490228T3+10357081509p5T4+58098p10T5+p15T6 1 + 58098 T + 10357081509 T^{2} + 373431668490228 T^{3} + 10357081509 p^{5} T^{4} + 58098 p^{10} T^{5} + p^{15} T^{6}
89S4×C2S_4\times C_2 119614T+11485034871T231522212744708T3+11485034871p5T419614p10T5+p15T6 1 - 19614 T + 11485034871 T^{2} - 31522212744708 T^{3} + 11485034871 p^{5} T^{4} - 19614 p^{10} T^{5} + p^{15} T^{6}
97S4×C2S_4\times C_2 187078T+14634643503T21303843661262292T3+14634643503p5T487078p10T5+p15T6 1 - 87078 T + 14634643503 T^{2} - 1303843661262292 T^{3} + 14634643503 p^{5} T^{4} - 87078 p^{10} T^{5} + p^{15} T^{6}
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   L(s)=p j=16(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.765670026982283328815422666282, −8.091084382420908790824892733140, −7.968868778022150678404246625404, −7.76159839654417269503164052501, −7.33961416325794839205208621935, −7.25715286287775521038886666802, −6.85741662499994260570537028487, −6.32779923722187225633952688426, −6.06721758565053115164935135011, −5.80399389139785005565484171032, −5.47222337540423347793952400417, −5.15976472861249016692667227020, −4.54005019225721683972820529774, −4.52782093436346387701863180333, −4.42700831901146259938861051574, −4.07525763361489831936597206476, −3.24190532139064035248684307828, −3.14965351235742196323327859547, −3.02730064186136903564756910865, −2.29995135908343013614979313403, −2.22063973322207767431042394291, −1.99113021213989685235262906358, −1.52453463465547112685543993256, −0.846448305910342452938626606605, −0.50783733117930365376203155698, 0.50783733117930365376203155698, 0.846448305910342452938626606605, 1.52453463465547112685543993256, 1.99113021213989685235262906358, 2.22063973322207767431042394291, 2.29995135908343013614979313403, 3.02730064186136903564756910865, 3.14965351235742196323327859547, 3.24190532139064035248684307828, 4.07525763361489831936597206476, 4.42700831901146259938861051574, 4.52782093436346387701863180333, 4.54005019225721683972820529774, 5.15976472861249016692667227020, 5.47222337540423347793952400417, 5.80399389139785005565484171032, 6.06721758565053115164935135011, 6.32779923722187225633952688426, 6.85741662499994260570537028487, 7.25715286287775521038886666802, 7.33961416325794839205208621935, 7.76159839654417269503164052501, 7.968868778022150678404246625404, 8.091084382420908790824892733140, 8.765670026982283328815422666282

Graph of the ZZ-function along the critical line