Properties

Label 4-3332e2-1.1-c0e2-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 9-s − 2·13-s + 5·16-s − 2·17-s − 2·18-s + 2·25-s + 4·26-s − 6·32-s + 4·34-s + 3·36-s − 4·50-s − 6·52-s − 2·53-s + 7·64-s − 6·68-s − 4·72-s − 2·89-s + 6·100-s + 4·101-s + 8·104-s + 4·106-s − 2·117-s + 121-s + 127-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 9-s − 2·13-s + 5·16-s − 2·17-s − 2·18-s + 2·25-s + 4·26-s − 6·32-s + 4·34-s + 3·36-s − 4·50-s − 6·52-s − 2·53-s + 7·64-s − 6·68-s − 4·72-s − 2·89-s + 6·100-s + 4·101-s + 8·104-s + 4·106-s − 2·117-s + 121-s + 127-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

\(L(\frac{1}{2})\) \(\approx\) \(0.3907989100\)
\(L(\frac12)\) \(\approx\) \(0.3907989100\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 - T^{2} + T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2^2$ \( 1 - T^{2} + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−8.902171554130202618911694926166, −8.836511413520278058370010727934, −8.299576845134420884616515967567, −7.933317147152552473405151975889, −7.45507297238138286722127694965, −7.23348364605972382626051290849, −6.92857563484196563808035938517, −6.69149825291659222536667894203, −6.27744887887884551330707218521, −5.86221188537596506521788798599, −5.16527165361072307011411089017, −4.73206168258376016760797597991, −4.56821751285816804460011112535, −3.84131042076333239670533691248, −2.99710921713900790186056301790, −2.94083115754563720068588777951, −2.21260851307020926107644850563, −1.98330077646091093478117271983, −1.36709598565501862331938183107, −0.51974453057723587216119713507, 0.51974453057723587216119713507, 1.36709598565501862331938183107, 1.98330077646091093478117271983, 2.21260851307020926107644850563, 2.94083115754563720068588777951, 2.99710921713900790186056301790, 3.84131042076333239670533691248, 4.56821751285816804460011112535, 4.73206168258376016760797597991, 5.16527165361072307011411089017, 5.86221188537596506521788798599, 6.27744887887884551330707218521, 6.69149825291659222536667894203, 6.92857563484196563808035938517, 7.23348364605972382626051290849, 7.45507297238138286722127694965, 7.933317147152552473405151975889, 8.299576845134420884616515967567, 8.836511413520278058370010727934, 8.902171554130202618911694926166

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