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Properties

Label 6-825e3-1.1-c1e3-0-3
Degree 66
Conductor 561515625561515625
Sign 11
Analytic cond. 285.886285.886
Root an. cond. 2.566642.56664
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 3·3-s + 4·4-s − 9·6-s + 8·7-s + 4·8-s + 6·9-s − 3·11-s − 12·12-s + 6·13-s + 24·14-s + 3·16-s + 4·17-s + 18·18-s − 2·19-s − 24·21-s − 9·22-s + 12·23-s − 12·24-s + 18·26-s − 10·27-s + 32·28-s − 8·29-s + 8·31-s − 32-s + 9·33-s + 12·34-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 2·4-s − 3.67·6-s + 3.02·7-s + 1.41·8-s + 2·9-s − 0.904·11-s − 3.46·12-s + 1.66·13-s + 6.41·14-s + 3/4·16-s + 0.970·17-s + 4.24·18-s − 0.458·19-s − 5.23·21-s − 1.91·22-s + 2.50·23-s − 2.44·24-s + 3.53·26-s − 1.92·27-s + 6.04·28-s − 1.48·29-s + 1.43·31-s − 0.176·32-s + 1.56·33-s + 2.05·34-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅113)s/2 ΓC(s)3 L(s)=( Λ(2−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(2-s)\end{aligned}
Λ(s)=((33⋅56⋅113)s/2 ΓC(s+1/2)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+1/2)^{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: 11
Analytic conductor: 285.886285.886
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 33⋅56⋅113, ( :1/2,1/2,1/2), 1)(6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 8.5441487648.544148764
L(12)L(\frac12) ≈\approx 8.5441487648.544148764
L(32)L(\frac{3}{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+T)3 ( 1 + T )^{3}
5 1 1
11C1C_1 (1+T)3 ( 1 + T )^{3}
good2S4×C2S_4\times C_2 1−3T+5T2−7T3+5pT4−3p2T5+p3T6 1 - 3 T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1−8T+37T2−116T3+37pT4−8p2T5+p3T6 1 - 8 T + 37 T^{2} - 116 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−6T+11T2−8T3+11pT4−6p2T5+p3T6 1 - 6 T + 11 T^{2} - 8 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−4T+23T2−20T3+23pT4−4p2T5+p3T6 1 - 4 T + 23 T^{2} - 20 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1+2T+5T2−108T3+5pT4+2p2T5+p3T6 1 + 2 T + 5 T^{2} - 108 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
23C2C_2 (1−4T+pT2)3 ( 1 - 4 T + p T^{2} )^{3}
29S4×C2S_4\times C_2 1+8T+3pT2+432T3+3p2T4+8p2T5+p3T6 1 + 8 T + 3 p T^{2} + 432 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−8T+101T2−480T3+101pT4−8p2T5+p3T6 1 - 8 T + 101 T^{2} - 480 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1−4T+95T2−264T3+95pT4−4p2T5+p3T6 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+8T+3pT2+624T3+3p2T4+8p2T5+p3T6 1 + 8 T + 3 p T^{2} + 624 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−8T+145T2−692T3+145pT4−8p2T5+p3T6 1 - 8 T + 145 T^{2} - 692 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−8T+125T2−592T3+125pT4−8p2T5+p3T6 1 - 8 T + 125 T^{2} - 592 T^{3} + 125 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+8T+127T2+576T3+127pT4+8p2T5+p3T6 1 + 8 T + 127 T^{2} + 576 T^{3} + 127 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+8T+113T2+1024T3+113pT4+8p2T5+p3T6 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−2T+131T2−204T3+131pT4−2p2T5+p3T6 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−12T+185T2−1288T3+185pT4−12p2T5+p3T6 1 - 12 T + 185 T^{2} - 1288 T^{3} + 185 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+12T+245T2+1688T3+245pT4+12p2T5+p3T6 1 + 12 T + 245 T^{2} + 1688 T^{3} + 245 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−18T+279T2−2536T3+279pT4−18p2T5+p3T6 1 - 18 T + 279 T^{2} - 2536 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+6T+233T2+940T3+233pT4+6p2T5+p3T6 1 + 6 T + 233 T^{2} + 940 T^{3} + 233 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+2T+245T2+328T3+245pT4+2p2T5+p3T6 1 + 2 T + 245 T^{2} + 328 T^{3} + 245 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1−2T+255T2−348T3+255pT4−2p2T5+p3T6 1 - 2 T + 255 T^{2} - 348 T^{3} + 255 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+8T+259T2+1424T3+259pT4+8p2T5+p3T6 1 + 8 T + 259 T^{2} + 1424 T^{3} + 259 p T^{4} + 8 p^{2} T^{5} + p^{3} 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

−9.168836514074333158517830630843, −8.535774714872927885589435672188, −8.511226800400000433009649511041, −8.042429170907351404348630653450, −7.978377094133140898193965544623, −7.47731373788340706321215657623, −7.34550030273298068337234858901, −6.96569658306558056074072397097, −6.47542481197842590998376621368, −6.38739929905039578222873250661, −5.81889204724798485412087037558, −5.58032556971060338129616761729, −5.39471893138383905450016926700, −5.07815310897846657421056372475, −4.98651758517345396645537036177, −4.67666615259883987453194053627, −4.28930424590553349006082575792, −4.16631700215969944884114739038, −3.79848399739394893566659218010, −3.16781990644841169311502821444, −2.88147475175510917268599223343, −2.08486401237106693669829180021, −1.75765100409066135275359600174, −1.16785437323250681236653867749, −0.963447407139270182895892958408, 0.963447407139270182895892958408, 1.16785437323250681236653867749, 1.75765100409066135275359600174, 2.08486401237106693669829180021, 2.88147475175510917268599223343, 3.16781990644841169311502821444, 3.79848399739394893566659218010, 4.16631700215969944884114739038, 4.28930424590553349006082575792, 4.67666615259883987453194053627, 4.98651758517345396645537036177, 5.07815310897846657421056372475, 5.39471893138383905450016926700, 5.58032556971060338129616761729, 5.81889204724798485412087037558, 6.38739929905039578222873250661, 6.47542481197842590998376621368, 6.96569658306558056074072397097, 7.34550030273298068337234858901, 7.47731373788340706321215657623, 7.978377094133140898193965544623, 8.042429170907351404348630653450, 8.511226800400000433009649511041, 8.535774714872927885589435672188, 9.168836514074333158517830630843

Graph of the ZZ-function along the critical line