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Properties

Label 6-825e3-1.1-c3e3-0-3
Degree 66
Conductor 561515625561515625
Sign −1-1
Analytic cond. 115334.115334.
Root an. cond. 6.976866.97686
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 33

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 9·3-s − 8·4-s − 9·6-s + 16·7-s − 12·8-s + 54·9-s + 33·11-s + 72·12-s + 45·13-s + 16·14-s − 7·16-s − 58·17-s + 54·18-s − 169·19-s − 144·21-s + 33·22-s + 155·23-s + 108·24-s + 45·26-s − 270·27-s − 128·28-s − 277·29-s − 173·31-s + 53·32-s − 297·33-s − 58·34-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.73·3-s − 4-s − 0.612·6-s + 0.863·7-s − 0.530·8-s + 2·9-s + 0.904·11-s + 1.73·12-s + 0.960·13-s + 0.305·14-s − 0.109·16-s − 0.827·17-s + 0.707·18-s − 2.04·19-s − 1.49·21-s + 0.319·22-s + 1.40·23-s + 0.918·24-s + 0.339·26-s − 1.92·27-s − 0.863·28-s − 1.77·29-s − 1.00·31-s + 0.292·32-s − 1.56·33-s − 0.292·34-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅113)s/2 ΓC(s)3 L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}
Λ(s)=((33⋅56⋅113)s/2 ΓC(s+3/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 33⋅56⋅1133^{3} \cdot 5^{6} \cdot 11^{3}
Sign: −1-1
Analytic conductor: 115334.115334.
Root analytic conductor: 6.976866.97686
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 33⋅56⋅113, ( :3/2,3/2,3/2), −1)(6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C1C_1 (1+pT)3 ( 1 + p T )^{3}
5 1 1
11C1C_1 (1−pT)3 ( 1 - p T )^{3}
good2S4×C2S_4\times C_2 1−T+9T2−5T3+9p3T4−p6T5+p9T6 1 - T + 9 T^{2} - 5 T^{3} + 9 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6}
7S4×C2S_4\times C_2 1−16T+856T2−11150T3+856p3T4−16p6T5+p9T6 1 - 16 T + 856 T^{2} - 11150 T^{3} + 856 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6}
13S4×C2S_4\times C_2 1−45T+6871T2−192794T3+6871p3T4−45p6T5+p9T6 1 - 45 T + 6871 T^{2} - 192794 T^{3} + 6871 p^{3} T^{4} - 45 p^{6} T^{5} + p^{9} T^{6}
17S4×C2S_4\times C_2 1+58T+15434T2+572224T3+15434p3T4+58p6T5+p9T6 1 + 58 T + 15434 T^{2} + 572224 T^{3} + 15434 p^{3} T^{4} + 58 p^{6} T^{5} + p^{9} T^{6}
19S4×C2S_4\times C_2 1+169T+28098T2+2360147T3+28098p3T4+169p6T5+p9T6 1 + 169 T + 28098 T^{2} + 2360147 T^{3} + 28098 p^{3} T^{4} + 169 p^{6} T^{5} + p^{9} T^{6}
23S4×C2S_4\times C_2 1−155T+41944T2−3812131T3+41944p3T4−155p6T5+p9T6 1 - 155 T + 41944 T^{2} - 3812131 T^{3} + 41944 p^{3} T^{4} - 155 p^{6} T^{5} + p^{9} T^{6}
29S4×C2S_4\times C_2 1+277T+68543T2+12078682T3+68543p3T4+277p6T5+p9T6 1 + 277 T + 68543 T^{2} + 12078682 T^{3} + 68543 p^{3} T^{4} + 277 p^{6} T^{5} + p^{9} T^{6}
31S4×C2S_4\times C_2 1+173T+42261T2+6587230T3+42261p3T4+173p6T5+p9T6 1 + 173 T + 42261 T^{2} + 6587230 T^{3} + 42261 p^{3} T^{4} + 173 p^{6} T^{5} + p^{9} T^{6}
37S4×C2S_4\times C_2 1−60T+3724T2+9069062T3+3724p3T4−60p6T5+p9T6 1 - 60 T + 3724 T^{2} + 9069062 T^{3} + 3724 p^{3} T^{4} - 60 p^{6} T^{5} + p^{9} T^{6}
41S4×C2S_4\times C_2 1−44T+133902T2−3107666T3+133902p3T4−44p6T5+p9T6 1 - 44 T + 133902 T^{2} - 3107666 T^{3} + 133902 p^{3} T^{4} - 44 p^{6} T^{5} + p^{9} T^{6}
43S4×C2S_4\times C_2 1+109T+228873T2+16964786T3+228873p3T4+109p6T5+p9T6 1 + 109 T + 228873 T^{2} + 16964786 T^{3} + 228873 p^{3} T^{4} + 109 p^{6} T^{5} + p^{9} T^{6}
47S4×C2S_4\times C_2 1−270T+208054T2−52560216T3+208054p3T4−270p6T5+p9T6 1 - 270 T + 208054 T^{2} - 52560216 T^{3} + 208054 p^{3} T^{4} - 270 p^{6} T^{5} + p^{9} T^{6}
53S4×C2S_4\times C_2 1+148T+385579T2+33949544T3+385579p3T4+148p6T5+p9T6 1 + 148 T + 385579 T^{2} + 33949544 T^{3} + 385579 p^{3} T^{4} + 148 p^{6} T^{5} + p^{9} T^{6}
59S4×C2S_4\times C_2 1+684T+535726T2+196887298T3+535726p3T4+684p6T5+p9T6 1 + 684 T + 535726 T^{2} + 196887298 T^{3} + 535726 p^{3} T^{4} + 684 p^{6} T^{5} + p^{9} T^{6}
61S4×C2S_4\times C_2 1+1038T+631795T2+333327140T3+631795p3T4+1038p6T5+p9T6 1 + 1038 T + 631795 T^{2} + 333327140 T^{3} + 631795 p^{3} T^{4} + 1038 p^{6} T^{5} + p^{9} T^{6}
67S4×C2S_4\times C_2 1−314T+854861T2−187279700T3+854861p3T4−314p6T5+p9T6 1 - 314 T + 854861 T^{2} - 187279700 T^{3} + 854861 p^{3} T^{4} - 314 p^{6} T^{5} + p^{9} T^{6}
71S4×C2S_4\times C_2 1+1459T+1463380T2+997438583T3+1463380p3T4+1459p6T5+p9T6 1 + 1459 T + 1463380 T^{2} + 997438583 T^{3} + 1463380 p^{3} T^{4} + 1459 p^{6} T^{5} + p^{9} T^{6}
73S4×C2S_4\times C_2 1+1170T+1393615T2+839920876T3+1393615p3T4+1170p6T5+p9T6 1 + 1170 T + 1393615 T^{2} + 839920876 T^{3} + 1393615 p^{3} T^{4} + 1170 p^{6} T^{5} + p^{9} T^{6}
79S4×C2S_4\times C_2 1+506T+1295660T2+507308816T3+1295660p3T4+506p6T5+p9T6 1 + 506 T + 1295660 T^{2} + 507308816 T^{3} + 1295660 p^{3} T^{4} + 506 p^{6} T^{5} + p^{9} T^{6}
83S4×C2S_4\times C_2 1+347T+1599297T2+390027914T3+1599297p3T4+347p6T5+p9T6 1 + 347 T + 1599297 T^{2} + 390027914 T^{3} + 1599297 p^{3} T^{4} + 347 p^{6} T^{5} + p^{9} T^{6}
89S4×C2S_4\times C_2 1+607T+1367067T2+593095898T3+1367067p3T4+607p6T5+p9T6 1 + 607 T + 1367067 T^{2} + 593095898 T^{3} + 1367067 p^{3} T^{4} + 607 p^{6} T^{5} + p^{9} T^{6}
97S4×C2S_4\times C_2 1−1263T+3146554T2−2315882599T3+3146554p3T4−1263p6T5+p9T6 1 - 1263 T + 3146554 T^{2} - 2315882599 T^{3} + 3146554 p^{3} T^{4} - 1263 p^{6} T^{5} + p^{9} T^{6}
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   L(s)=∏p ∏j=16(1−αj,p p−s)−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

−9.263293876261939526661978520545, −8.702915205585140090904192747814, −8.664123869522520366132567728921, −8.622194468202238909872669325819, −7.80966644406917050810989327776, −7.73926583350674272645103976609, −7.21397965371771272436779538667, −7.17278476585545262622482414384, −6.55932097755422708933437199250, −6.47012140906186325521478136931, −6.02987606153651054039243844245, −5.95008204989237871528680858803, −5.68899766698085835265853581391, −4.98944415978559613841897480702, −4.94185744590564265955608826780, −4.73523686464525354459439630888, −4.22345054335914501648876979701, −4.10040012031390303175985092269, −3.96320387692851659190594900196, −3.23054795752020912331568928382, −2.97029458858689015338525211301, −2.13242561146774600131448802271, −1.77524093157596250946077326926, −1.26226370906427648111550413283, −1.24704046627505110969373102320, 0, 0, 0, 1.24704046627505110969373102320, 1.26226370906427648111550413283, 1.77524093157596250946077326926, 2.13242561146774600131448802271, 2.97029458858689015338525211301, 3.23054795752020912331568928382, 3.96320387692851659190594900196, 4.10040012031390303175985092269, 4.22345054335914501648876979701, 4.73523686464525354459439630888, 4.94185744590564265955608826780, 4.98944415978559613841897480702, 5.68899766698085835265853581391, 5.95008204989237871528680858803, 6.02987606153651054039243844245, 6.47012140906186325521478136931, 6.55932097755422708933437199250, 7.17278476585545262622482414384, 7.21397965371771272436779538667, 7.73926583350674272645103976609, 7.80966644406917050810989327776, 8.622194468202238909872669325819, 8.664123869522520366132567728921, 8.702915205585140090904192747814, 9.263293876261939526661978520545

Graph of the ZZ-function along the critical line