Properties

Label 4-3332e2-1.1-c1e2-0-7
Degree $4$
Conductor $11102224$
Sign $1$
Analytic cond. $707.887$
Root an. cond. $5.15811$
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  + 3·3-s + 3·5-s + 4·9-s + 2·11-s + 6·13-s + 9·15-s + 2·17-s + 12·19-s + 2·23-s + 6·27-s − 10·29-s + 17·31-s + 6·33-s + 18·39-s − 41-s − 5·43-s + 12·45-s − 20·47-s + 6·51-s + 5·53-s + 6·55-s + 36·57-s + 4·59-s + 5·61-s + 18·65-s + 9·67-s + 6·69-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s + 4/3·9-s + 0.603·11-s + 1.66·13-s + 2.32·15-s + 0.485·17-s + 2.75·19-s + 0.417·23-s + 1.15·27-s − 1.85·29-s + 3.05·31-s + 1.04·33-s + 2.88·39-s − 0.156·41-s − 0.762·43-s + 1.78·45-s − 2.91·47-s + 0.840·51-s + 0.686·53-s + 0.809·55-s + 4.76·57-s + 0.520·59-s + 0.640·61-s + 2.23·65-s + 1.09·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11102224\)    =    \(2^{4} \cdot 7^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(707.887\)
Root analytic conductor: \(5.15811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11102224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.84156361\)
\(L(\frac12)\) \(\approx\) \(10.84156361\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 17 T + 131 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
43$C_4$ \( 1 + 5 T + 89 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 5 T + 109 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 125 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 9 T + 85 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 15 T + 221 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.682387309683576626356959481357, −8.506688500161231004311711585035, −8.108448796978046193475231524049, −7.961124940640164801546031569083, −7.43376244387374215714960849448, −6.83359675553433149076650620813, −6.73764853677029265620042500565, −6.16257719436519209809635571122, −5.84620362911401061994813275465, −5.42672818410414175540176096598, −5.04154532832702039834195383205, −4.60398173680048519162805186593, −3.77203565043264282322181540255, −3.66158674754342727022811804463, −3.10613542763688161550250061349, −2.99898620483043452236770836242, −2.32490400693043135370915291419, −1.82186369562600974388642463745, −1.17417811731172735545052185636, −1.11457962680886830333445368701, 1.11457962680886830333445368701, 1.17417811731172735545052185636, 1.82186369562600974388642463745, 2.32490400693043135370915291419, 2.99898620483043452236770836242, 3.10613542763688161550250061349, 3.66158674754342727022811804463, 3.77203565043264282322181540255, 4.60398173680048519162805186593, 5.04154532832702039834195383205, 5.42672818410414175540176096598, 5.84620362911401061994813275465, 6.16257719436519209809635571122, 6.73764853677029265620042500565, 6.83359675553433149076650620813, 7.43376244387374215714960849448, 7.961124940640164801546031569083, 8.108448796978046193475231524049, 8.506688500161231004311711585035, 8.682387309683576626356959481357

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