Properties

Label 12-77e6-1.1-c1e6-0-1
Degree $12$
Conductor $208422380089$
Sign $1$
Analytic cond. $0.0540265$
Root an. cond. $0.784122$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s + 2·5-s + 2·7-s − 6·8-s + 5·9-s + 3·11-s + 12-s − 22·13-s + 2·15-s + 2·16-s + 3·17-s + 11·19-s + 2·20-s + 2·21-s − 12·23-s − 6·24-s + 8·25-s + 8·27-s + 2·28-s − 18·29-s + 3·31-s − 9·32-s + 3·33-s + 4·35-s + 5·36-s + 4·37-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.755·7-s − 2.12·8-s + 5/3·9-s + 0.904·11-s + 0.288·12-s − 6.10·13-s + 0.516·15-s + 1/2·16-s + 0.727·17-s + 2.52·19-s + 0.447·20-s + 0.436·21-s − 2.50·23-s − 1.22·24-s + 8/5·25-s + 1.53·27-s + 0.377·28-s − 3.34·29-s + 0.538·31-s − 1.59·32-s + 0.522·33-s + 0.676·35-s + 5/6·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(0.0540265\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{77} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8734733697\)
\(L(\frac12)\) \(\approx\) \(0.8734733697\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 2 T + 2 p T^{2} - 23 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11 \( ( 1 - T + T^{2} )^{3} \)
good2 \( 1 - T^{2} + 3 p T^{3} - T^{4} - 3 T^{5} + 23 T^{6} - 3 p T^{7} - p^{2} T^{8} + 3 p^{4} T^{9} - p^{4} T^{10} + p^{6} T^{12} \)
3 \( 1 - T - 4 T^{2} + T^{3} + 7 T^{4} + 2 p T^{5} - 7 p T^{6} + 2 p^{2} T^{7} + 7 p^{2} T^{8} + p^{3} T^{9} - 4 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T - 4 T^{2} + 14 T^{3} - 6 T^{4} - 2 p T^{5} + 11 p T^{6} - 2 p^{2} T^{7} - 6 p^{2} T^{8} + 14 p^{3} T^{9} - 4 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 11 T + 75 T^{2} + 321 T^{3} + 75 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 3 T - 40 T^{2} + 43 T^{3} + 1283 T^{4} - 524 T^{5} - 24567 T^{6} - 524 p T^{7} + 1283 p^{2} T^{8} + 43 p^{3} T^{9} - 40 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 11 T + 44 T^{2} - 125 T^{3} + 495 T^{4} + 22 p T^{5} - 647 p T^{6} + 22 p^{2} T^{7} + 495 p^{2} T^{8} - 125 p^{3} T^{9} + 44 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 32 T^{2} + 146 T^{3} + 2780 T^{4} + 11612 T^{5} + 18447 T^{6} + 11612 p T^{7} + 2780 p^{2} T^{8} + 146 p^{3} T^{9} + 32 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 9 T + 67 T^{2} + 469 T^{3} + 67 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 3 T - 40 T^{2} + 11 T^{3} + 645 T^{4} + 2360 T^{5} - 17257 T^{6} + 2360 p T^{7} + 645 p^{2} T^{8} + 11 p^{3} T^{9} - 40 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 4 T - 59 T^{2} - 12 T^{3} + 2274 T^{4} + 5924 T^{5} - 107987 T^{6} + 5924 p T^{7} + 2274 p^{2} T^{8} - 12 p^{3} T^{9} - 59 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 5 T + 43 T^{2} + 301 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 2 T + 104 T^{2} - 131 T^{3} + 104 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 3 T - 130 T^{2} + 133 T^{3} + 11963 T^{4} - 5654 T^{5} - 644697 T^{6} - 5654 p T^{7} + 11963 p^{2} T^{8} + 133 p^{3} T^{9} - 130 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 17 T + 56 T^{2} + 315 T^{3} + 10949 T^{4} + 52646 T^{5} - 68035 T^{6} + 52646 p T^{7} + 10949 p^{2} T^{8} + 315 p^{3} T^{9} + 56 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 8 T + 44 T^{2} + 918 T^{3} + 2252 T^{4} + 1388 T^{5} + 308015 T^{6} + 1388 p T^{7} + 2252 p^{2} T^{8} + 918 p^{3} T^{9} + 44 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 24 T + 221 T^{2} - 1912 T^{3} + 25074 T^{4} - 222784 T^{5} + 1539557 T^{6} - 222784 p T^{7} + 25074 p^{2} T^{8} - 1912 p^{3} T^{9} + 221 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 16 T - 13 T^{2} + 128 T^{3} + 18882 T^{4} - 91192 T^{5} - 485189 T^{6} - 91192 p T^{7} + 18882 p^{2} T^{8} + 128 p^{3} T^{9} - 13 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 7 T + 127 T^{2} - 575 T^{3} + 127 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 20 T + 156 T^{2} - 290 T^{3} - 4176 T^{4} + 45920 T^{5} - 432669 T^{6} + 45920 p T^{7} - 4176 p^{2} T^{8} - 290 p^{3} T^{9} + 156 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 3 T - 190 T^{2} - 69 T^{3} + 22881 T^{4} - 9336 T^{5} - 2087681 T^{6} - 9336 p T^{7} + 22881 p^{2} T^{8} - 69 p^{3} T^{9} - 190 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 11 T + 265 T^{2} + 1823 T^{3} + 265 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + T - 258 T^{2} - 91 T^{3} + 43855 T^{4} + 7144 T^{5} - 4537567 T^{6} + 7144 p T^{7} + 43855 p^{2} T^{8} - 91 p^{3} T^{9} - 258 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 9 T + 279 T^{2} - 1699 T^{3} + 279 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−8.311849014791559589021185830396, −7.967490264424783029488090657076, −7.67813894525452560083223214992, −7.50413124166115584482810759714, −7.46548999128718578251289408523, −7.32869327540054136508016920748, −6.99157384316780366550277920037, −6.97300884166097984311781674530, −6.55550942873089056883468716152, −6.17201802612164602826761736773, −6.09766773250978124093875935884, −5.84755281498972572103272440736, −5.20138593770132509944543673969, −5.18601066721870050876048389686, −5.01040394459861023201370972214, −5.00620061781217152127091036705, −4.73888022017137878603403898941, −3.96176796408239184367296603891, −3.70626388647952044536539493509, −3.63727125864432032688310576714, −3.00018392625285238886447387149, −2.65292920569784627899543212349, −2.52994888200094150951199022014, −1.94715995869985345784970239046, −1.82956938649959526708922619529, 1.82956938649959526708922619529, 1.94715995869985345784970239046, 2.52994888200094150951199022014, 2.65292920569784627899543212349, 3.00018392625285238886447387149, 3.63727125864432032688310576714, 3.70626388647952044536539493509, 3.96176796408239184367296603891, 4.73888022017137878603403898941, 5.00620061781217152127091036705, 5.01040394459861023201370972214, 5.18601066721870050876048389686, 5.20138593770132509944543673969, 5.84755281498972572103272440736, 6.09766773250978124093875935884, 6.17201802612164602826761736773, 6.55550942873089056883468716152, 6.97300884166097984311781674530, 6.99157384316780366550277920037, 7.32869327540054136508016920748, 7.46548999128718578251289408523, 7.50413124166115584482810759714, 7.67813894525452560083223214992, 7.967490264424783029488090657076, 8.311849014791559589021185830396

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.