Properties

Label 12-74e6-1.1-c1e6-0-0
Degree $12$
Conductor $164206490176$
Sign $1$
Analytic cond. $0.0425650$
Root an. cond. $0.768695$
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  − 3·2-s + 3·4-s − 5-s + 2·8-s + 9-s + 3·10-s + 8·11-s − 6·13-s − 9·16-s + 3·17-s − 3·18-s − 8·19-s − 3·20-s − 24·22-s + 16·23-s − 2·25-s + 18·26-s − 8·27-s − 6·29-s + 8·31-s + 9·32-s − 9·34-s + 3·36-s − 11·37-s + 24·38-s − 2·40-s + 7·41-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 0.447·5-s + 0.707·8-s + 1/3·9-s + 0.948·10-s + 2.41·11-s − 1.66·13-s − 9/4·16-s + 0.727·17-s − 0.707·18-s − 1.83·19-s − 0.670·20-s − 5.11·22-s + 3.33·23-s − 2/5·25-s + 3.53·26-s − 1.53·27-s − 1.11·29-s + 1.43·31-s + 1.59·32-s − 1.54·34-s + 1/2·36-s − 1.80·37-s + 3.89·38-s − 0.316·40-s + 1.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(0.0425650\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1876022586\)
\(L(\frac12)\) \(\approx\) \(0.1876022586\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
37 \( 1 + 11 T + 134 T^{2} + 815 T^{3} + 134 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( 1 - T^{2} + 8 T^{3} - 2 T^{4} - 4 T^{5} + 67 T^{6} - 4 p T^{7} - 2 p^{2} T^{8} + 8 p^{3} T^{9} - p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 + T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} + 67 T^{5} + 326 T^{6} + 67 p T^{7} + p^{4} T^{8} + 4 p^{4} T^{9} + 3 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 13 T^{2} - 8 T^{3} + 78 T^{4} + 52 T^{5} - 481 T^{6} + 52 p T^{7} + 78 p^{2} T^{8} - 8 p^{3} T^{9} - 13 p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - 4 T + 17 T^{2} - 40 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 - 5 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \)
17 \( 1 - 3 T - 13 T^{2} + 12 T^{3} - 7 T^{4} + 519 T^{5} + 518 T^{6} + 519 p T^{7} - 7 p^{2} T^{8} + 12 p^{3} T^{9} - 13 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 8 T + 15 T^{2} + 8 T^{3} - 66 T^{4} - 2600 T^{5} - 18141 T^{6} - 2600 p T^{7} - 66 p^{2} T^{8} + 8 p^{3} T^{9} + 15 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 - 8 T + 73 T^{2} - 356 T^{3} + 73 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 + 3 T + 82 T^{2} + 171 T^{3} + 82 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - 4 T + 69 T^{2} - 264 T^{3} + 69 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 7 T - 61 T^{2} + 372 T^{3} + 3593 T^{4} - 11437 T^{5} - 114586 T^{6} - 11437 p T^{7} + 3593 p^{2} T^{8} + 372 p^{3} T^{9} - 61 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 - 8 T + 109 T^{2} - 540 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( ( 1 + 16 T + 197 T^{2} + 1552 T^{3} + 197 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 + 22 T + 185 T^{2} + 1482 T^{3} + 17498 T^{4} + 126382 T^{5} + 692249 T^{6} + 126382 p T^{7} + 17498 p^{2} T^{8} + 1482 p^{3} T^{9} + 185 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 4 T - T^{2} + 468 T^{3} + 86 T^{4} + 2500 T^{5} + 378011 T^{6} + 2500 p T^{7} + 86 p^{2} T^{8} + 468 p^{3} T^{9} - p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 9 T - 121 T^{2} - 392 T^{3} + 18537 T^{4} + 34903 T^{5} - 1115458 T^{6} + 34903 p T^{7} + 18537 p^{2} T^{8} - 392 p^{3} T^{9} - 121 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 16 T - 9 T^{2} + 80 T^{3} + 17910 T^{4} - 76064 T^{5} - 587133 T^{6} - 76064 p T^{7} + 17910 p^{2} T^{8} + 80 p^{3} T^{9} - 9 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T - 109 T^{2} + 444 T^{3} + 25166 T^{4} - 69600 T^{5} - 1527121 T^{6} - 69600 p T^{7} + 25166 p^{2} T^{8} + 444 p^{3} T^{9} - 109 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 - 2 T + 199 T^{2} - 300 T^{3} + 199 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 4 T - 149 T^{2} + 892 T^{3} + 11086 T^{4} - 48724 T^{5} - 662377 T^{6} - 48724 p T^{7} + 11086 p^{2} T^{8} + 892 p^{3} T^{9} - 149 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 4 T - 169 T^{2} - 852 T^{3} + 15686 T^{4} + 51976 T^{5} - 1173709 T^{6} + 51976 p T^{7} + 15686 p^{2} T^{8} - 852 p^{3} T^{9} - 169 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T - 37 T^{2} - 1044 T^{3} - 6055 T^{4} + 6507 T^{5} + 637910 T^{6} + 6507 p T^{7} - 6055 p^{2} T^{8} - 1044 p^{3} T^{9} - 37 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 13 T + 318 T^{2} - 2529 T^{3} + 318 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−8.317900127744862278149754496246, −8.225214218751247309642443753991, −7.949830522824951245566425204696, −7.81874030242604022224583846277, −7.50971158780724160299084532504, −7.36414970961526755457077853257, −7.32487999167103792420946246589, −6.67466518249955352430153828515, −6.67264220177622341610529616069, −6.60238576215399886777773952153, −6.43184687677280413923488605373, −5.91975186596407756035892670039, −5.52058422804668733797722400848, −5.40924915975832073907732961599, −4.87238793835004083883923872346, −4.64489223391926068959373998079, −4.50928289791509269796628937971, −4.44228348321657210616630175821, −3.72835324663060223169942872204, −3.50830827326973275751603700338, −3.40537620758945713518790439201, −2.75299454404612744510344940506, −2.01186820963435793697241308306, −1.83336077981812360404894545468, −1.04701229545292129802348670970, 1.04701229545292129802348670970, 1.83336077981812360404894545468, 2.01186820963435793697241308306, 2.75299454404612744510344940506, 3.40537620758945713518790439201, 3.50830827326973275751603700338, 3.72835324663060223169942872204, 4.44228348321657210616630175821, 4.50928289791509269796628937971, 4.64489223391926068959373998079, 4.87238793835004083883923872346, 5.40924915975832073907732961599, 5.52058422804668733797722400848, 5.91975186596407756035892670039, 6.43184687677280413923488605373, 6.60238576215399886777773952153, 6.67264220177622341610529616069, 6.67466518249955352430153828515, 7.32487999167103792420946246589, 7.36414970961526755457077853257, 7.50971158780724160299084532504, 7.81874030242604022224583846277, 7.949830522824951245566425204696, 8.225214218751247309642443753991, 8.317900127744862278149754496246

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

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