Properties

Label 4-3332e2-1.1-c0e2-0-5
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 + 16-s + 2·17-s − 2·20-s + 2·25-s + 2·29-s − 2·37-s − 2·41-s + 2·61-s − 64-s − 2·68-s − 2·73-s + 2·80-s − 81-s + 4·85-s + 2·97-s − 2·100-s − 2·109-s − 2·113-s − 2·116-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + ⋯
L(s)  = 1  − 4-s + 2·5-s + 16-s + 2·17-s − 2·20-s + 2·25-s + 2·29-s − 2·37-s − 2·41-s + 2·61-s − 64-s − 2·68-s − 2·73-s + 2·80-s − 81-s + 4·85-s + 2·97-s − 2·100-s − 2·109-s − 2·113-s − 2·116-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-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: induced by $\chi_{3332} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11102224,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.844794739\)
\(L(\frac12)\) \(\approx\) \(1.844794739\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−8.988317510774996668541736067437, −8.697110020316634563007092859195, −8.218461130640259915870701265046, −8.202334424155764257956765858584, −7.50575887639937940949158821573, −6.97185243947968735825637832093, −6.81343652147486035026172025044, −6.21055863605209663476274700992, −5.94295530227994839951083086231, −5.40497688836295314121031880538, −5.37516842637644558425263657366, −4.90901771012926610417118480275, −4.60127189344037187120258141629, −3.71589776636210439291140395299, −3.63283968892761488034441673367, −2.82000327023422200456686663783, −2.76639974321406096970467182040, −1.70607229695763933127210759749, −1.58963277607082267870183528224, −0.894378655439629435642484357070, 0.894378655439629435642484357070, 1.58963277607082267870183528224, 1.70607229695763933127210759749, 2.76639974321406096970467182040, 2.82000327023422200456686663783, 3.63283968892761488034441673367, 3.71589776636210439291140395299, 4.60127189344037187120258141629, 4.90901771012926610417118480275, 5.37516842637644558425263657366, 5.40497688836295314121031880538, 5.94295530227994839951083086231, 6.21055863605209663476274700992, 6.81343652147486035026172025044, 6.97185243947968735825637832093, 7.50575887639937940949158821573, 8.202334424155764257956765858584, 8.218461130640259915870701265046, 8.697110020316634563007092859195, 8.988317510774996668541736067437

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