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Properties

Label 12-722e6-1.1-c1e6-0-8
Degree $12$
Conductor $1.417\times 10^{17}$
Sign $1$
Analytic cond. $36718.5$
Root an. cond. $2.40108$
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 − 6·5-s + 12·7-s + 2·8-s + 6·9-s + 18·10-s + 12·11-s + 6·13-s − 36·14-s − 9·16-s − 3·17-s − 18·18-s − 18·20-s − 36·22-s + 27·25-s − 18·26-s − 2·27-s + 36·28-s − 12·29-s − 12·31-s + 9·32-s + 9·34-s − 72·35-s + 18·36-s − 12·37-s − 12·40-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 2.68·5-s + 4.53·7-s + 0.707·8-s + 2·9-s + 5.69·10-s + 3.61·11-s + 1.66·13-s − 9.62·14-s − 9/4·16-s − 0.727·17-s − 4.24·18-s − 4.02·20-s − 7.67·22-s + 27/5·25-s − 3.53·26-s − 0.384·27-s + 6.80·28-s − 2.22·29-s − 2.15·31-s + 1.59·32-s + 1.54·34-s − 12.1·35-s + 3·36-s − 1.97·37-s − 1.89·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(36718.5\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.644524128\)
\(L(\frac12)\) \(\approx\) \(1.644524128\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
19 \( 1 \)
good3 \( 1 - 2 p T^{2} + 2 T^{3} + 2 p^{2} T^{4} - 2 p T^{5} - 53 T^{6} - 2 p^{2} T^{7} + 2 p^{4} T^{8} + 2 p^{3} T^{9} - 2 p^{5} T^{10} + p^{6} T^{12} \)
5 \( ( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3} \)
7 \( ( 1 - 6 T + 3 p T^{2} - 60 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 6 T + 36 T^{2} - 113 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 6 T - 3 T^{2} + 62 T^{3} + 126 T^{4} - 54 p T^{5} + 441 T^{6} - 54 p^{2} T^{7} + 126 p^{2} T^{8} + 62 p^{3} T^{9} - 3 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 3 T + 3 T^{2} + 36 T^{3} - 39 T^{4} + 201 T^{5} + 6694 T^{6} + 201 p T^{7} - 39 p^{2} T^{8} + 36 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 57 T^{2} - 16 T^{3} + 1938 T^{4} + 456 T^{5} - 50329 T^{6} + 456 p T^{7} + 1938 p^{2} T^{8} - 16 p^{3} T^{9} - 57 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 12 T + 45 T^{2} + 100 T^{3} + 402 T^{4} - 4956 T^{5} - 60619 T^{6} - 4956 p T^{7} + 402 p^{2} T^{8} + 100 p^{3} T^{9} + 45 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 6 T + 69 T^{2} + 380 T^{3} + 69 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 6 T + 87 T^{2} + 308 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 120 T^{2} - 2 T^{3} + 9480 T^{4} + 120 T^{5} - 452557 T^{6} + 120 p T^{7} + 9480 p^{2} T^{8} - 2 p^{3} T^{9} - 120 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 9 T - 63 T^{2} - 218 T^{3} + 7731 T^{4} + 14193 T^{5} - 309354 T^{6} + 14193 p T^{7} + 7731 p^{2} T^{8} - 218 p^{3} T^{9} - 63 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T - 33 T^{2} + 122 T^{3} - 90 T^{4} + 7170 T^{5} - 23821 T^{6} + 7170 p T^{7} - 90 p^{2} T^{8} + 122 p^{3} T^{9} - 33 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 75 T^{2} + 272 T^{3} + 1650 T^{4} - 10200 T^{5} + 18125 T^{6} - 10200 p T^{7} + 1650 p^{2} T^{8} + 272 p^{3} T^{9} - 75 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 - 6 T - 132 T^{2} + 414 T^{3} + 14046 T^{4} - 21390 T^{5} - 874253 T^{6} - 21390 p T^{7} + 14046 p^{2} T^{8} + 414 p^{3} T^{9} - 132 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 18 T + 45 T^{2} + 326 T^{3} + 18558 T^{4} + 109386 T^{5} + 13161 T^{6} + 109386 p T^{7} + 18558 p^{2} T^{8} + 326 p^{3} T^{9} + 45 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 18 T + 42 T^{2} - 90 T^{3} + 15756 T^{4} - 103644 T^{5} + 34031 T^{6} - 103644 p T^{7} + 15756 p^{2} T^{8} - 90 p^{3} T^{9} + 42 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T - 81 T^{2} + 404 T^{3} + 16026 T^{4} - 26292 T^{5} - 1195753 T^{6} - 26292 p T^{7} + 16026 p^{2} T^{8} + 404 p^{3} T^{9} - 81 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 12 T - 12 T^{2} - 1518 T^{3} - 7392 T^{4} + 55056 T^{5} + 1107155 T^{6} + 55056 p T^{7} - 7392 p^{2} T^{8} - 1518 p^{3} T^{9} - 12 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 18 T - 9 T^{2} - 34 T^{3} + 30942 T^{4} - 157626 T^{5} - 915837 T^{6} - 157626 p T^{7} + 30942 p^{2} T^{8} - 34 p^{3} T^{9} - 9 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 6 T + 222 T^{2} - 945 T^{3} + 222 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 9 T - 69 T^{2} + 108 T^{3} + 3561 T^{4} - 75861 T^{5} - 848522 T^{6} - 75861 p T^{7} + 3561 p^{2} T^{8} + 108 p^{3} T^{9} - 69 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 288 T^{2} - 2 T^{3} + 55008 T^{4} + 288 T^{5} - 6220221 T^{6} + 288 p T^{7} + 55008 p^{2} T^{8} - 2 p^{3} T^{9} - 288 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.29688124048360640207864093804, −5.27942919757935495160727419481, −5.24622549836487148853682757370, −5.16621122909524892779286257992, −4.65825449046446000792908430817, −4.54385684052327578401465725399, −4.41329018991040694891099078527, −4.37716491116094457053321543287, −4.12226092526031727896370259244, −4.11997899241078236663987698467, −3.93594139998928469755898451284, −3.59686696304683171210286921918, −3.52746389480352806646931811035, −3.47085753583239408720677616630, −3.43438602398601780251692827293, −2.57867777261666134303316123665, −2.43828171141272599897389947806, −1.90977854449687938000546942481, −1.72722035898901814293837348049, −1.57704780748974334864361594147, −1.57472160829391486460520325753, −1.44170510358132283446788503226, −1.13655825220601057003077281154, −0.903886194473310057110824857436, −0.37733124772711937784309671886, 0.37733124772711937784309671886, 0.903886194473310057110824857436, 1.13655825220601057003077281154, 1.44170510358132283446788503226, 1.57472160829391486460520325753, 1.57704780748974334864361594147, 1.72722035898901814293837348049, 1.90977854449687938000546942481, 2.43828171141272599897389947806, 2.57867777261666134303316123665, 3.43438602398601780251692827293, 3.47085753583239408720677616630, 3.52746389480352806646931811035, 3.59686696304683171210286921918, 3.93594139998928469755898451284, 4.11997899241078236663987698467, 4.12226092526031727896370259244, 4.37716491116094457053321543287, 4.41329018991040694891099078527, 4.54385684052327578401465725399, 4.65825449046446000792908430817, 5.16621122909524892779286257992, 5.24622549836487148853682757370, 5.27942919757935495160727419481, 5.29688124048360640207864093804

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

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