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Properties

Label 12-1183e6-1.1-c1e6-0-2
Degree $12$
Conductor $2.741\times 10^{18}$
Sign $1$
Analytic cond. $710511.$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s − 2·5-s + 8·6-s + 6·7-s + 3·8-s − 9-s + 4·10-s − 8·11-s − 12·14-s + 8·15-s − 6·16-s − 23·17-s + 2·18-s + 13·19-s − 24·21-s + 16·22-s − 18·23-s − 12·24-s − 18·25-s + 26·27-s − 15·29-s − 16·30-s − 3·31-s + 2·32-s + 32·33-s + 46·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s − 0.894·5-s + 3.26·6-s + 2.26·7-s + 1.06·8-s − 1/3·9-s + 1.26·10-s − 2.41·11-s − 3.20·14-s + 2.06·15-s − 3/2·16-s − 5.57·17-s + 0.471·18-s + 2.98·19-s − 5.23·21-s + 3.41·22-s − 3.75·23-s − 2.44·24-s − 3.59·25-s + 5.00·27-s − 2.78·29-s − 2.92·30-s − 0.538·31-s + 0.353·32-s + 5.57·33-s + 7.88·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(710511.\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 7^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 - T )^{6} \)
13 \( 1 \)
good2 \( 1 + p T + p^{2} T^{2} + 5 T^{3} + 5 p T^{4} + 9 p T^{5} + 31 T^{6} + 9 p^{2} T^{7} + 5 p^{3} T^{8} + 5 p^{3} T^{9} + p^{6} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 + 4 T + 17 T^{2} + 46 T^{3} + 122 T^{4} + 245 T^{5} + 481 T^{6} + 245 p T^{7} + 122 p^{2} T^{8} + 46 p^{3} T^{9} + 17 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 2 T + 22 T^{2} + 7 p T^{3} + 229 T^{4} + 303 T^{5} + 1447 T^{6} + 303 p T^{7} + 229 p^{2} T^{8} + 7 p^{4} T^{9} + 22 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 8 T + 50 T^{2} + 243 T^{3} + 1083 T^{4} + 4383 T^{5} + 15277 T^{6} + 4383 p T^{7} + 1083 p^{2} T^{8} + 243 p^{3} T^{9} + 50 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 23 T + 292 T^{2} + 2540 T^{3} + 16936 T^{4} + 91115 T^{5} + 409523 T^{6} + 91115 p T^{7} + 16936 p^{2} T^{8} + 2540 p^{3} T^{9} + 292 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 13 T + 161 T^{2} - 1215 T^{3} + 8665 T^{4} - 45671 T^{5} + 227327 T^{6} - 45671 p T^{7} + 8665 p^{2} T^{8} - 1215 p^{3} T^{9} + 161 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 18 T + 235 T^{2} + 2091 T^{3} + 15661 T^{4} + 94020 T^{5} + 495523 T^{6} + 94020 p T^{7} + 15661 p^{2} T^{8} + 2091 p^{3} T^{9} + 235 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 183 T^{2} + 1577 T^{3} + 12663 T^{4} + 80831 T^{5} + 478995 T^{6} + 80831 p T^{7} + 12663 p^{2} T^{8} + 1577 p^{3} T^{9} + 183 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 29 T^{2} - 99 T^{3} + 1365 T^{4} + 1247 T^{5} + 67011 T^{6} + 1247 p T^{7} + 1365 p^{2} T^{8} - 99 p^{3} T^{9} + 29 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 13 T + 227 T^{2} - 2287 T^{3} + 21261 T^{4} - 165075 T^{5} + 1053087 T^{6} - 165075 p T^{7} + 21261 p^{2} T^{8} - 2287 p^{3} T^{9} + 227 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 131 T^{2} - 292 T^{3} + 6158 T^{4} - 3460 T^{5} + 202879 T^{6} - 3460 p T^{7} + 6158 p^{2} T^{8} - 292 p^{3} T^{9} + 131 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 18 T + 311 T^{2} + 3496 T^{3} + 35744 T^{4} + 284555 T^{5} + 2083101 T^{6} + 284555 p T^{7} + 35744 p^{2} T^{8} + 3496 p^{3} T^{9} + 311 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 16 T + 260 T^{2} - 2471 T^{3} + 24701 T^{4} - 179041 T^{5} + 1411093 T^{6} - 179041 p T^{7} + 24701 p^{2} T^{8} - 2471 p^{3} T^{9} + 260 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 25 T + 429 T^{2} + 5407 T^{3} + 58624 T^{4} + 523460 T^{5} + 4125969 T^{6} + 523460 p T^{7} + 58624 p^{2} T^{8} + 5407 p^{3} T^{9} + 429 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 18 T + 231 T^{2} + 2285 T^{3} + 20723 T^{4} + 164956 T^{5} + 1340761 T^{6} + 164956 p T^{7} + 20723 p^{2} T^{8} + 2285 p^{3} T^{9} + 231 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 16 T + 311 T^{2} - 3792 T^{3} + 44666 T^{4} - 408293 T^{5} + 3575773 T^{6} - 408293 p T^{7} + 44666 p^{2} T^{8} - 3792 p^{3} T^{9} + 311 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 16 T + 296 T^{2} + 2748 T^{3} + 28371 T^{4} + 190491 T^{5} + 1772499 T^{6} + 190491 p T^{7} + 28371 p^{2} T^{8} + 2748 p^{3} T^{9} + 296 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 25 T + 450 T^{2} + 5633 T^{3} + 66618 T^{4} + 665639 T^{5} + 6202577 T^{6} + 665639 p T^{7} + 66618 p^{2} T^{8} + 5633 p^{3} T^{9} + 450 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 5 T + 262 T^{2} - 1812 T^{3} + 33912 T^{4} - 262035 T^{5} + 2891423 T^{6} - 262035 p T^{7} + 33912 p^{2} T^{8} - 1812 p^{3} T^{9} + 262 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 2 T + 307 T^{2} - 689 T^{3} + 48091 T^{4} - 95350 T^{5} + 4742205 T^{6} - 95350 p T^{7} + 48091 p^{2} T^{8} - 689 p^{3} T^{9} + 307 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 7 T + 226 T^{2} - 2485 T^{3} + 35590 T^{4} - 306775 T^{5} + 3896483 T^{6} - 306775 p T^{7} + 35590 p^{2} T^{8} - 2485 p^{3} T^{9} + 226 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 10 T + 292 T^{2} - 2764 T^{3} + 48371 T^{4} - 414955 T^{5} + 5171855 T^{6} - 414955 p T^{7} + 48371 p^{2} T^{8} - 2764 p^{3} T^{9} + 292 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 5 T + 286 T^{2} - 1431 T^{3} + 39905 T^{4} - 196126 T^{5} + 4141037 T^{6} - 196126 p T^{7} + 39905 p^{2} T^{8} - 1431 p^{3} T^{9} + 286 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.64698405096049395237399495019, −5.59352013421617940925415256787, −5.26194837770239625972476826135, −5.19394652366583432137449245271, −5.07311972145462707238561508820, −4.90285014430311740308920988579, −4.80547974805746724007742504023, −4.53978455244479268350370192070, −4.30962428559482472550915904021, −4.24652080262011934969699892630, −4.10943052805080200519435660120, −4.07490714585619872008789571773, −3.90931170615413242494446834323, −3.45187646142047637092227272616, −3.36198429017099241016754447383, −3.21497644317474005675824965950, −2.62496170229204047211364004208, −2.60239125648269313273355711268, −2.44950928692679634503499833963, −2.17627762296277189559130789171, −2.14908700499738215939212614929, −1.93886454377728794391497652841, −1.69442318639432463619272445310, −1.38819368418339663339080442496, −1.13138530065947532124486780832, 0, 0, 0, 0, 0, 0, 1.13138530065947532124486780832, 1.38819368418339663339080442496, 1.69442318639432463619272445310, 1.93886454377728794391497652841, 2.14908700499738215939212614929, 2.17627762296277189559130789171, 2.44950928692679634503499833963, 2.60239125648269313273355711268, 2.62496170229204047211364004208, 3.21497644317474005675824965950, 3.36198429017099241016754447383, 3.45187646142047637092227272616, 3.90931170615413242494446834323, 4.07490714585619872008789571773, 4.10943052805080200519435660120, 4.24652080262011934969699892630, 4.30962428559482472550915904021, 4.53978455244479268350370192070, 4.80547974805746724007742504023, 4.90285014430311740308920988579, 5.07311972145462707238561508820, 5.19394652366583432137449245271, 5.26194837770239625972476826135, 5.59352013421617940925415256787, 5.64698405096049395237399495019

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

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