Properties

Label 4-164e2-1.1-c0e2-0-0
Degree $4$
Conductor $26896$
Sign $1$
Analytic cond. $0.00669887$
Root an. cond. $0.286088$
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  − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 2·25-s − 6·32-s + 2·41-s + 4·50-s − 4·61-s + 7·64-s − 81-s − 4·82-s − 6·100-s + 8·122-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 6·164-s + 167-s + 2·169-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 2·25-s − 6·32-s + 2·41-s + 4·50-s − 4·61-s + 7·64-s − 81-s − 4·82-s − 6·100-s + 8·122-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 6·164-s + 167-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26896\)    =    \(2^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(0.00669887\)
Root analytic conductor: \(0.286088\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{164} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 26896,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2002762706\)
\(L(\frac12)\) \(\approx\) \(0.2002762706\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
41$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$ \( ( 1 + T )^{4} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−13.18763867837712168221678951560, −12.51833864361520564569950340750, −12.16056558853819033271782428387, −11.59109040205698406505356516171, −11.25219882261673457397142875486, −10.64515854163204002054397390352, −10.30712458699540265229593530298, −9.694866627355286219347224837793, −9.187105519113440642461821141702, −9.024266535468335828680798604443, −7.958130159799043255838239927925, −7.955392826552363289980170655329, −7.33632915986505954677530283474, −6.69921581546472461321751958786, −5.85609162058118679786749317933, −5.83779551634800273249445863948, −4.40272053803656526702806000032, −3.38262324183870129332137983046, −2.53413337667629057470742593665, −1.61066445217681730691510386296, 1.61066445217681730691510386296, 2.53413337667629057470742593665, 3.38262324183870129332137983046, 4.40272053803656526702806000032, 5.83779551634800273249445863948, 5.85609162058118679786749317933, 6.69921581546472461321751958786, 7.33632915986505954677530283474, 7.955392826552363289980170655329, 7.958130159799043255838239927925, 9.024266535468335828680798604443, 9.187105519113440642461821141702, 9.694866627355286219347224837793, 10.30712458699540265229593530298, 10.64515854163204002054397390352, 11.25219882261673457397142875486, 11.59109040205698406505356516171, 12.16056558853819033271782428387, 12.51833864361520564569950340750, 13.18763867837712168221678951560

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