Properties

Label 4-26e2-1.1-c1e2-0-1
Degree $4$
Conductor $676$
Sign $1$
Analytic cond. $0.0431023$
Root an. cond. $0.455643$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s − 3·9-s + 2·12-s + 4·13-s + 16-s + 6·17-s − 12·23-s + 25-s + 14·27-s + 3·36-s − 8·39-s − 2·43-s − 2·48-s + 5·49-s − 12·51-s − 4·52-s − 12·53-s − 16·61-s − 64-s − 6·68-s + 24·69-s − 2·75-s + 20·79-s − 4·81-s + 12·92-s − 100-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 9-s + 0.577·12-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 2.50·23-s + 1/5·25-s + 2.69·27-s + 1/2·36-s − 1.28·39-s − 0.304·43-s − 0.288·48-s + 5/7·49-s − 1.68·51-s − 0.554·52-s − 1.64·53-s − 2.04·61-s − 1/8·64-s − 0.727·68-s + 2.88·69-s − 0.230·75-s + 2.25·79-s − 4/9·81-s + 1.25·92-s − 0.0999·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.0431023\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 676,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3168060727\)
\(L(\frac12)\) \(\approx\) \(0.3168060727\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.69518615041136556702863141543, −17.36882624358461094367934412757, −16.49082166238566739681079280254, −16.44997075291564349503830075472, −15.59041563034695015036545603647, −14.62874621167701846399328933010, −14.00419847866366404346413565835, −13.77322524224678752979164692856, −12.56955955660686972306929715682, −12.06840841229674551327782820165, −11.58320108030543861280167783156, −10.78604730673694910604982598109, −10.24097835334179608955390309155, −9.273169857449595050678731052098, −8.380034989348270758343219773095, −7.81580411016520448253038462683, −6.10272337101188958807676622402, −6.01632195233972618102175746207, −4.94931671065091800429494720066, −3.49748557902694879901443144890, 3.49748557902694879901443144890, 4.94931671065091800429494720066, 6.01632195233972618102175746207, 6.10272337101188958807676622402, 7.81580411016520448253038462683, 8.380034989348270758343219773095, 9.273169857449595050678731052098, 10.24097835334179608955390309155, 10.78604730673694910604982598109, 11.58320108030543861280167783156, 12.06840841229674551327782820165, 12.56955955660686972306929715682, 13.77322524224678752979164692856, 14.00419847866366404346413565835, 14.62874621167701846399328933010, 15.59041563034695015036545603647, 16.44997075291564349503830075472, 16.49082166238566739681079280254, 17.36882624358461094367934412757, 17.69518615041136556702863141543

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