Properties

Label 4-44e4-1.1-c0e2-0-0
Degree $4$
Conductor $3748096$
Sign $1$
Analytic cond. $0.933522$
Root an. cond. $0.982949$
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  + 2·5-s − 9-s + 25-s − 2·37-s − 2·45-s + 2·49-s + 4·53-s − 2·89-s + 2·97-s + 2·113-s − 2·125-s + ⋯
L(s)  = 1  + 2·5-s − 9-s + 25-s − 2·37-s − 2·45-s + 2·49-s + 4·53-s − 2·89-s + 2·97-s + 2·113-s − 2·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3748096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3748096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3748096\)    =    \(2^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(0.933522\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3748096,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.620398660\)
\(L(\frac12)\) \(\approx\) \(1.620398660\)
\(L(1)\) 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 \( 1 \)
11 \( 1 \)
good3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 + T + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
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$C_1$ \( ( 1 - T )^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )^{2} \)
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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

−9.640317268061211596020781495400, −9.148101785823214346316215680293, −8.825185252665606564260127816555, −8.456216113936814409569794857538, −8.370309324335241466508505510262, −7.34203700380208073842251413750, −7.28038330883140208100791869749, −6.85571113774531123733552260913, −6.18106632074979565840448163790, −5.95759143686629620108376772735, −5.59415716688718267131208884364, −5.37655199759165305566032061663, −4.96238703781524110973354693141, −4.17709879599193285017811065570, −3.80539463535116529056997276897, −3.16935562769062200964573307684, −2.59667525607659379049587207602, −2.16014133613123843159751795174, −1.86641200668441300789558030836, −0.954593618280825410273854867640, 0.954593618280825410273854867640, 1.86641200668441300789558030836, 2.16014133613123843159751795174, 2.59667525607659379049587207602, 3.16935562769062200964573307684, 3.80539463535116529056997276897, 4.17709879599193285017811065570, 4.96238703781524110973354693141, 5.37655199759165305566032061663, 5.59415716688718267131208884364, 5.95759143686629620108376772735, 6.18106632074979565840448163790, 6.85571113774531123733552260913, 7.28038330883140208100791869749, 7.34203700380208073842251413750, 8.370309324335241466508505510262, 8.456216113936814409569794857538, 8.825185252665606564260127816555, 9.148101785823214346316215680293, 9.640317268061211596020781495400

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