Properties

Label 12-825e6-1.1-c1e6-0-0
Degree $12$
Conductor $3.153\times 10^{17}$
Sign $1$
Analytic cond. $81730.9$
Root an. cond. $2.56664$
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  + 4-s − 3·9-s + 6·11-s + 2·16-s − 16·19-s + 20·29-s + 16·31-s − 3·36-s − 28·41-s + 6·44-s + 10·49-s − 24·59-s − 12·61-s + 6·64-s + 16·71-s − 16·76-s − 24·79-s + 6·81-s + 20·89-s − 18·99-s − 52·101-s + 12·109-s + 20·116-s + 21·121-s + 16·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1/2·4-s − 9-s + 1.80·11-s + 1/2·16-s − 3.67·19-s + 3.71·29-s + 2.87·31-s − 1/2·36-s − 4.37·41-s + 0.904·44-s + 10/7·49-s − 3.12·59-s − 1.53·61-s + 3/4·64-s + 1.89·71-s − 1.83·76-s − 2.70·79-s + 2/3·81-s + 2.11·89-s − 1.80·99-s − 5.17·101-s + 1.14·109-s + 1.85·116-s + 1.90·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{12} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(81730.9\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{12} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.770462390\)
\(L(\frac12)\) \(\approx\) \(2.770462390\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{3} \)
5 \( 1 \)
11 \( ( 1 - T )^{6} \)
good2 \( 1 - T^{2} - T^{4} - 3 T^{6} - p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 - 10 T^{2} + 95 T^{4} - 780 T^{6} + 95 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 - 8 T + 7 T^{2} + 64 T^{3} + 7 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )( 1 + 8 T + 7 T^{2} - 64 T^{3} + 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} ) \)
17 \( 1 + 6 T^{2} + 431 T^{4} + 5908 T^{6} + 431 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 8 T + 41 T^{2} + 144 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 10 T^{2} + 255 T^{4} - 9100 T^{6} + 255 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 12 T + p T^{2} )^{3}( 1 + 12 T + p T^{2} )^{3} \)
41 \( ( 1 + 14 T + 167 T^{2} + 1156 T^{3} + 167 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 82 T^{2} + 7063 T^{4} - 303196 T^{6} + 7063 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 154 T^{2} + 12143 T^{4} - 652332 T^{6} + 12143 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 178 T^{2} + 15063 T^{4} - 894364 T^{6} + 15063 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + 12 T + 161 T^{2} + 1096 T^{3} + 161 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 290 T^{2} + 40135 T^{4} - 3371900 T^{6} + 40135 p^{2} T^{8} - 290 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 8 T + 181 T^{2} - 1008 T^{3} + 181 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 250 T^{2} + 34687 T^{4} - 3059500 T^{6} + 34687 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 12 T + 173 T^{2} + 1096 T^{3} + 173 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 258 T^{2} + 38055 T^{4} - 3905724 T^{6} + 38055 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 10 T + 215 T^{2} - 1580 T^{3} + 215 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 314 T^{2} + 48463 T^{4} - 5318252 T^{6} + 48463 p^{2} T^{8} - 314 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.52707032904482241445914892365, −5.11637270973882560114949791046, −5.08179408184155358246922501546, −4.84477184131292186806258237260, −4.76248189017050633318029523059, −4.62968548506691528379060573417, −4.42055673436722627067979442307, −4.26505844248580017466678776321, −4.18583498199407337787606462020, −3.94791924414931075329544765749, −3.62709873265667025876242237144, −3.58844764605204203956616777997, −3.26364735823302301800308967992, −3.19805980919521265372365458022, −2.83793424940657830106505447380, −2.66738533295421041852669406306, −2.60164682178265468323678504723, −2.47068811344536406231443898767, −2.15449222198825069803858356654, −1.71452847983268594848801885341, −1.61313959608575948932764737755, −1.31990427721858131701545632652, −1.28445986406779272986953938203, −0.61630173451540393108419477284, −0.32592891858079128549973764519, 0.32592891858079128549973764519, 0.61630173451540393108419477284, 1.28445986406779272986953938203, 1.31990427721858131701545632652, 1.61313959608575948932764737755, 1.71452847983268594848801885341, 2.15449222198825069803858356654, 2.47068811344536406231443898767, 2.60164682178265468323678504723, 2.66738533295421041852669406306, 2.83793424940657830106505447380, 3.19805980919521265372365458022, 3.26364735823302301800308967992, 3.58844764605204203956616777997, 3.62709873265667025876242237144, 3.94791924414931075329544765749, 4.18583498199407337787606462020, 4.26505844248580017466678776321, 4.42055673436722627067979442307, 4.62968548506691528379060573417, 4.76248189017050633318029523059, 4.84477184131292186806258237260, 5.08179408184155358246922501546, 5.11637270973882560114949791046, 5.52707032904482241445914892365

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

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