Properties

Label 4-2156e2-1.1-c1e2-0-0
Degree $4$
Conductor $4648336$
Sign $1$
Analytic cond. $296.381$
Root an. cond. $4.14918$
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-s + 3·5-s + 3·9-s + 11-s − 8·13-s − 3·15-s − 6·17-s − 8·19-s + 3·23-s + 5·25-s − 8·27-s − 5·31-s − 33-s + 37-s + 8·39-s − 20·43-s + 9·45-s + 6·51-s + 6·53-s + 3·55-s + 8·57-s − 3·59-s + 4·61-s − 24·65-s + 67-s − 3·69-s + 30·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 9-s + 0.301·11-s − 2.21·13-s − 0.774·15-s − 1.45·17-s − 1.83·19-s + 0.625·23-s + 25-s − 1.53·27-s − 0.898·31-s − 0.174·33-s + 0.164·37-s + 1.28·39-s − 3.04·43-s + 1.34·45-s + 0.840·51-s + 0.824·53-s + 0.404·55-s + 1.05·57-s − 0.390·59-s + 0.512·61-s − 2.97·65-s + 0.122·67-s − 0.361·69-s + 3.56·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4648336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4648336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4648336\)    =    \(2^{4} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(296.381\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4648336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8324429761\)
\(L(\frac12)\) \(\approx\) \(0.8324429761\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 - T + T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−9.646607645759054046137974914087, −8.839880730912391657347691186684, −8.747584011157596826577554754734, −7.905181756568216937940783618826, −7.88037807443422884732617997125, −7.00795988353557304507964240780, −6.79439039161101389638796102348, −6.63157028160009452573068518083, −6.36601718689926981296252211529, −5.44816726269419656633045030770, −5.42619355643597567989477200556, −4.77073721108431445045230278798, −4.76770504851687484346015379312, −3.91133395522680153929712870254, −3.77229161854014070862065872863, −2.64195737578491255764831793282, −2.38237176205914225597149531827, −1.92070140309929650413065459521, −1.53448885903603392648822590898, −0.30791476009999026734898523567, 0.30791476009999026734898523567, 1.53448885903603392648822590898, 1.92070140309929650413065459521, 2.38237176205914225597149531827, 2.64195737578491255764831793282, 3.77229161854014070862065872863, 3.91133395522680153929712870254, 4.76770504851687484346015379312, 4.77073721108431445045230278798, 5.42619355643597567989477200556, 5.44816726269419656633045030770, 6.36601718689926981296252211529, 6.63157028160009452573068518083, 6.79439039161101389638796102348, 7.00795988353557304507964240780, 7.88037807443422884732617997125, 7.905181756568216937940783618826, 8.747584011157596826577554754734, 8.839880730912391657347691186684, 9.646607645759054046137974914087

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