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Properties

Label 4-3332e2-1.1-c0e2-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8856150876\)
\(L(\frac12)\) \(\approx\) \(0.8856150876\)
\(L(1)\) 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

−8.790390094214946818624091041978, −8.548549548274563564144906219624, −8.274342717562745769936631149975, −8.098908181856115567616957731689, −7.48358014587181380502623646717, −7.41986238014984211437602090300, −6.84314331405163844154461594750, −6.15038956349376653971524218849, −6.00542313895195228226626409227, −5.69325487277978777709354685503, −5.26087062239580847145273039016, −4.40871171970737961750501582965, −4.13088693244731173451210324301, −3.94905446137667590140928639264, −3.85246361645720185831238690570, −3.11611928451943545657526856909, −2.96896215101482196034074936860, −1.77697439988707306132956663952, −1.15790871148051782974946018628, −0.71656054501944727655211663883, 0.71656054501944727655211663883, 1.15790871148051782974946018628, 1.77697439988707306132956663952, 2.96896215101482196034074936860, 3.11611928451943545657526856909, 3.85246361645720185831238690570, 3.94905446137667590140928639264, 4.13088693244731173451210324301, 4.40871171970737961750501582965, 5.26087062239580847145273039016, 5.69325487277978777709354685503, 6.00542313895195228226626409227, 6.15038956349376653971524218849, 6.84314331405163844154461594750, 7.41986238014984211437602090300, 7.48358014587181380502623646717, 8.098908181856115567616957731689, 8.274342717562745769936631149975, 8.548549548274563564144906219624, 8.790390094214946818624091041978

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