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Properties

Label 12-8112e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.849\times 10^{23}$
Sign $1$
Analytic cond. $7.38635\times 10^{10}$
Root an. cond. $8.04826$
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  − 6·3-s + 5-s + 5·7-s + 21·9-s + 6·11-s − 6·15-s + 9·17-s − 7·19-s − 30·21-s − 12·23-s − 10·25-s − 56·27-s + 7·29-s + 11·31-s − 36·33-s + 5·35-s + 6·37-s + 13·41-s − 15·43-s + 21·45-s + 9·47-s − 2·49-s − 54·51-s + 22·53-s + 6·55-s + 42·57-s + 7·59-s + ⋯
L(s)  = 1  − 3.46·3-s + 0.447·5-s + 1.88·7-s + 7·9-s + 1.80·11-s − 1.54·15-s + 2.18·17-s − 1.60·19-s − 6.54·21-s − 2.50·23-s − 2·25-s − 10.7·27-s + 1.29·29-s + 1.97·31-s − 6.26·33-s + 0.845·35-s + 0.986·37-s + 2.03·41-s − 2.28·43-s + 3.13·45-s + 1.31·47-s − 2/7·49-s − 7.56·51-s + 3.02·53-s + 0.809·55-s + 5.56·57-s + 0.911·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(7.38635\times 10^{10}\)
Root analytic conductor: \(8.04826\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.10609933\)
\(L(\frac12)\) \(\approx\) \(13.10609933\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T )^{6} \)
13 \( 1 \)
good5 \( 1 - T + 11 T^{2} - 8 T^{3} + 98 T^{4} - 66 T^{5} + 553 T^{6} - 66 p T^{7} + 98 p^{2} T^{8} - 8 p^{3} T^{9} + 11 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 5 T + 27 T^{2} - 106 T^{3} + 54 p T^{4} - 160 p T^{5} + 463 p T^{6} - 160 p^{2} T^{7} + 54 p^{3} T^{8} - 106 p^{3} T^{9} + 27 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T + 43 T^{2} - 212 T^{3} + 929 T^{4} - 3478 T^{5} + 12603 T^{6} - 3478 p T^{7} + 929 p^{2} T^{8} - 212 p^{3} T^{9} + 43 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 9 T + 114 T^{2} - 716 T^{3} + 5091 T^{4} - 23603 T^{5} + 117036 T^{6} - 23603 p T^{7} + 5091 p^{2} T^{8} - 716 p^{3} T^{9} + 114 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 7 T + 94 T^{2} + 504 T^{3} + 3907 T^{4} + 16989 T^{5} + 95148 T^{6} + 16989 p T^{7} + 3907 p^{2} T^{8} + 504 p^{3} T^{9} + 94 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 123 T^{2} + 807 T^{3} + 5631 T^{4} + 30803 T^{5} + 170082 T^{6} + 30803 p T^{7} + 5631 p^{2} T^{8} + 807 p^{3} T^{9} + 123 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 7 T + 3 p T^{2} - 308 T^{3} + 3534 T^{4} - 13160 T^{5} + 134813 T^{6} - 13160 p T^{7} + 3534 p^{2} T^{8} - 308 p^{3} T^{9} + 3 p^{5} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 11 T + 203 T^{2} - 50 p T^{3} + 15928 T^{4} - 90560 T^{5} + 656651 T^{6} - 90560 p T^{7} + 15928 p^{2} T^{8} - 50 p^{4} T^{9} + 203 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 6 T + 133 T^{2} - 427 T^{3} + 6955 T^{4} - 9763 T^{5} + 252486 T^{6} - 9763 p T^{7} + 6955 p^{2} T^{8} - 427 p^{3} T^{9} + 133 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 13 T + 130 T^{2} - 992 T^{3} + 8541 T^{4} - 49751 T^{5} + 322416 T^{6} - 49751 p T^{7} + 8541 p^{2} T^{8} - 992 p^{3} T^{9} + 130 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 141 T^{2} + 1430 T^{3} + 14903 T^{4} + 106255 T^{5} + 669262 T^{6} + 106255 p T^{7} + 14903 p^{2} T^{8} + 1430 p^{3} T^{9} + 141 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 9 T + 2 p T^{2} + 114 T^{3} + 643 T^{4} + 14423 T^{5} + 140348 T^{6} + 14423 p T^{7} + 643 p^{2} T^{8} + 114 p^{3} T^{9} + 2 p^{5} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 11 T + 183 T^{2} - 1179 T^{3} + 183 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 7 T + 204 T^{2} - 238 T^{3} + 10077 T^{4} + 87493 T^{5} + 180708 T^{6} + 87493 p T^{7} + 10077 p^{2} T^{8} - 238 p^{3} T^{9} + 204 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 25 T + 440 T^{2} - 5900 T^{3} + 66737 T^{4} - 638391 T^{5} + 5343900 T^{6} - 638391 p T^{7} + 66737 p^{2} T^{8} - 5900 p^{3} T^{9} + 440 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 5 T + 226 T^{2} - 354 T^{3} + 12789 T^{4} - 190155 T^{5} + 293336 T^{6} - 190155 p T^{7} + 12789 p^{2} T^{8} - 354 p^{3} T^{9} + 226 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 8 T + 281 T^{2} + 2669 T^{3} + 38549 T^{4} + 365241 T^{5} + 3333546 T^{6} + 365241 p T^{7} + 38549 p^{2} T^{8} + 2669 p^{3} T^{9} + 281 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 15 T + 3 p T^{2} - 1456 T^{3} + 9232 T^{4} + 19750 T^{5} - 99129 T^{6} + 19750 p T^{7} + 9232 p^{2} T^{8} - 1456 p^{3} T^{9} + 3 p^{5} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 14 T + 145 T^{2} + 616 T^{3} + 11799 T^{4} + 137088 T^{5} + 1824577 T^{6} + 137088 p T^{7} + 11799 p^{2} T^{8} + 616 p^{3} T^{9} + 145 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 13 T + 221 T^{2} - 1524 T^{3} + 20630 T^{4} - 159630 T^{5} + 2122165 T^{6} - 159630 p T^{7} + 20630 p^{2} T^{8} - 1524 p^{3} T^{9} + 221 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 33 T + 710 T^{2} - 10824 T^{3} + 141665 T^{4} - 1595947 T^{5} + 16164832 T^{6} - 1595947 p T^{7} + 141665 p^{2} T^{8} - 10824 p^{3} T^{9} + 710 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 50 T + 1348 T^{2} - 25615 T^{3} + 388179 T^{4} - 4903428 T^{5} + 52476469 T^{6} - 4903428 p T^{7} + 388179 p^{2} T^{8} - 25615 p^{3} T^{9} + 1348 p^{4} T^{10} - 50 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.10914348474358909398772971008, −3.73161456050311054173001359807, −3.70095658655244850289083061854, −3.68602756071497931789419075835, −3.61969712176480753263369174105, −3.58895016828834836526438840647, −3.46948667523266157644638476313, −2.90848529907190023941713171698, −2.85662742903286406135610667005, −2.54864892775825147151617413064, −2.51929469416784074306513715974, −2.37230400376298737884160258640, −2.23287489208638259475199482169, −1.99067946678711648392753642403, −1.87542465146466671374429171755, −1.73926565836427560469243693822, −1.57780239749543315642213550192, −1.57546140117840278411647577923, −1.34649632986667081806190102378, −1.02046372436874436399691816702, −0.814530689350820264150388090541, −0.78865785705694722522238445336, −0.53639090342328516579900039334, −0.51399357558329414844235897212, −0.45312167905259150373295350368, 0.45312167905259150373295350368, 0.51399357558329414844235897212, 0.53639090342328516579900039334, 0.78865785705694722522238445336, 0.814530689350820264150388090541, 1.02046372436874436399691816702, 1.34649632986667081806190102378, 1.57546140117840278411647577923, 1.57780239749543315642213550192, 1.73926565836427560469243693822, 1.87542465146466671374429171755, 1.99067946678711648392753642403, 2.23287489208638259475199482169, 2.37230400376298737884160258640, 2.51929469416784074306513715974, 2.54864892775825147151617413064, 2.85662742903286406135610667005, 2.90848529907190023941713171698, 3.46948667523266157644638476313, 3.58895016828834836526438840647, 3.61969712176480753263369174105, 3.68602756071497931789419075835, 3.70095658655244850289083061854, 3.73161456050311054173001359807, 4.10914348474358909398772971008

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

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