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Properties

Label 4-3332e2-1.1-c0e2-0-2
Degree $4$
Conductor $11102224$
Sign $1$
Analytic cond. $2.76518$
Root an. cond. $1.28952$
Motivic weight $0$
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  − 4-s + 2·5-s − 4·13-s + 16-s − 2·20-s + 2·25-s − 2·29-s + 2·37-s − 2·41-s + 4·52-s − 2·61-s − 64-s − 8·65-s + 2·73-s + 2·80-s − 81-s + 4·89-s + 2·97-s − 2·100-s − 4·101-s + 2·109-s − 2·113-s + 2·116-s + 2·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 4-s + 2·5-s − 4·13-s + 16-s − 2·20-s + 2·25-s − 2·29-s + 2·37-s − 2·41-s + 4·52-s − 2·61-s − 64-s − 8·65-s + 2·73-s + 2·80-s − 81-s + 4·89-s + 2·97-s − 2·100-s − 4·101-s + 2·109-s − 2·113-s + 2·116-s + 2·125-s + 127-s + 131-s + 137-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(11102224\)    =    \(2^{4} \cdot 7^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2.76518\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11102224,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9443279954\)
\(L(\frac12)\) \(\approx\) \(0.9443279954\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
7 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$ \( ( 1 + T )^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$ \( ( 1 - T )^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−9.293129452313635962850210808535, −8.936886535696815813808001273117, −8.119807390076456274170699016941, −7.79903300281211056905051076760, −7.62692154896981139874941901936, −7.19126732277324764819272267579, −6.61810494245726834085852894239, −6.48409094057237388477506045423, −5.73626471775706529259774505670, −5.53950092468412514274425129532, −5.12865094577231853148438181420, −5.01755391402236332033188650721, −4.42878025722632013149202268664, −4.24200820661517750504143262776, −3.26447926822368127229734667222, −3.00120623832427226630588368322, −2.38674720264578613763938481049, −1.97716610158020560763087206113, −1.77070181713590931447225921994, −0.55044963260828938184894677577, 0.55044963260828938184894677577, 1.77070181713590931447225921994, 1.97716610158020560763087206113, 2.38674720264578613763938481049, 3.00120623832427226630588368322, 3.26447926822368127229734667222, 4.24200820661517750504143262776, 4.42878025722632013149202268664, 5.01755391402236332033188650721, 5.12865094577231853148438181420, 5.53950092468412514274425129532, 5.73626471775706529259774505670, 6.48409094057237388477506045423, 6.61810494245726834085852894239, 7.19126732277324764819272267579, 7.62692154896981139874941901936, 7.79903300281211056905051076760, 8.119807390076456274170699016941, 8.936886535696815813808001273117, 9.293129452313635962850210808535

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