Properties

Label 12-63e12-1.1-c1e6-0-0
Degree $12$
Conductor $3.909\times 10^{21}$
Sign $1$
Analytic cond. $1.01332\times 10^{9}$
Root an. cond. $5.62962$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4-s − 2·8-s + 8·11-s + 3·16-s + 16·22-s + 4·23-s − 9·25-s + 22·29-s − 2·32-s − 6·37-s + 6·43-s − 8·44-s + 8·46-s − 18·50-s + 28·53-s + 44·58-s − 14·64-s + 38·71-s − 12·74-s − 6·79-s + 12·86-s − 16·88-s − 4·92-s + 9·100-s + 56·106-s + 26·107-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 0.707·8-s + 2.41·11-s + 3/4·16-s + 3.41·22-s + 0.834·23-s − 9/5·25-s + 4.08·29-s − 0.353·32-s − 0.986·37-s + 0.914·43-s − 1.20·44-s + 1.17·46-s − 2.54·50-s + 3.84·53-s + 5.77·58-s − 7/4·64-s + 4.50·71-s − 1.39·74-s − 0.675·79-s + 1.29·86-s − 1.70·88-s − 0.417·92-s + 9/10·100-s + 5.43·106-s + 2.51·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{24} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.01332\times 10^{9}\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3969} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{24} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(27.78135937\)
\(L(\frac12)\) \(\approx\) \(27.78135937\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - T + p T^{2} - 3 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 + 9 T^{2} + 63 T^{4} + 349 T^{6} + 63 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - 4 T + 32 T^{2} - 87 T^{3} + 32 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 3 p T^{2} + 66 p T^{4} + 13439 T^{6} + 66 p^{3} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} \)
17 \( 1 + 18 T^{2} + 216 T^{4} + 2797 T^{6} + 216 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 39 T^{2} + 1011 T^{4} + 20009 T^{6} + 1011 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 - 2 T + 44 T^{2} - 33 T^{3} + 44 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 - 11 T + 101 T^{2} - 549 T^{3} + 101 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 57 T^{2} + 3927 T^{4} + 3731 p T^{6} + 3927 p^{2} T^{8} + 57 p^{4} T^{10} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 87 T^{2} + 249 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 84 T^{2} + 4722 T^{4} + 236077 T^{6} + 4722 p^{2} T^{8} + 84 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 3 T + 105 T^{2} - 285 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 99 T^{2} + 9315 T^{4} + 451393 T^{6} + 9315 p^{2} T^{8} + 99 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 - 14 T + 170 T^{2} - 1221 T^{3} + 170 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 171 T^{2} + 18477 T^{4} + 1250773 T^{6} + 18477 p^{2} T^{8} + 171 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 + 102 T^{2} + 12918 T^{4} + 712865 T^{6} + 12918 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 + 90 T^{2} + 353 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 19 T + 329 T^{2} - 2925 T^{3} + 329 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 363 T^{2} + 59331 T^{4} + 5567537 T^{6} + 59331 p^{2} T^{8} + 363 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 3 T + 159 T^{2} + 367 T^{3} + 159 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 270 T^{2} + 41004 T^{4} + 4002649 T^{6} + 41004 p^{2} T^{8} + 270 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 288 T^{2} + 36990 T^{4} + 3427693 T^{6} + 36990 p^{2} T^{8} + 288 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 471 T^{2} + 100065 T^{4} + 12364877 T^{6} + 100065 p^{2} T^{8} + 471 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.31987869036341559765755536140, −4.00458516795650871789295242193, −3.99676714357992567559688925848, −3.95997914762161076373903823865, −3.91710892117932815420682975278, −3.88181932460863600953722406991, −3.70405601715430039924753512822, −3.49045364410729629812740329400, −3.28709052250769518220578943218, −2.98976702205115896092848855908, −2.96101010513588786207001810354, −2.81843840527343720068408973709, −2.57474905930389913073495002766, −2.55123928727730680987026449706, −2.32564849613131091216071532632, −1.93658296258068489834258856543, −1.89422952642985197485481230319, −1.84702942276003742528666019902, −1.47193594344431941904935909953, −1.44306510287697189816581458684, −0.943066418682506317245108975028, −0.908011565394066867124676276782, −0.849636426600035289535070129240, −0.57463188208252772248240942128, −0.35615733911652037239925243447, 0.35615733911652037239925243447, 0.57463188208252772248240942128, 0.849636426600035289535070129240, 0.908011565394066867124676276782, 0.943066418682506317245108975028, 1.44306510287697189816581458684, 1.47193594344431941904935909953, 1.84702942276003742528666019902, 1.89422952642985197485481230319, 1.93658296258068489834258856543, 2.32564849613131091216071532632, 2.55123928727730680987026449706, 2.57474905930389913073495002766, 2.81843840527343720068408973709, 2.96101010513588786207001810354, 2.98976702205115896092848855908, 3.28709052250769518220578943218, 3.49045364410729629812740329400, 3.70405601715430039924753512822, 3.88181932460863600953722406991, 3.91710892117932815420682975278, 3.95997914762161076373903823865, 3.99676714357992567559688925848, 4.00458516795650871789295242193, 4.31987869036341559765755536140

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

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