Properties

Label 4-2e8-1.1-c1e2-0-0
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $0.0163227$
Root an. cond. $0.357436$
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  − 2·2-s − 2·3-s + 2·4-s − 2·5-s + 4·6-s + 2·9-s + 4·10-s + 2·11-s − 4·12-s − 2·13-s + 4·15-s − 4·16-s − 4·17-s − 4·18-s + 6·19-s − 4·20-s − 4·22-s + 2·25-s + 4·26-s − 6·27-s + 6·29-s − 8·30-s − 16·31-s + 8·32-s − 4·33-s + 8·34-s + 4·36-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s + 2/3·9-s + 1.26·10-s + 0.603·11-s − 1.15·12-s − 0.554·13-s + 1.03·15-s − 16-s − 0.970·17-s − 0.942·18-s + 1.37·19-s − 0.894·20-s − 0.852·22-s + 2/5·25-s + 0.784·26-s − 1.15·27-s + 1.11·29-s − 1.46·30-s − 2.87·31-s + 1.41·32-s − 0.696·33-s + 1.37·34-s + 2/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(0.0163227\)
Root analytic conductor: \(0.357436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{16} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 256,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1342091471\)
\(L(\frac12)\) \(\approx\) \(0.1342091471\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−19.49325330313906922783187166826, −18.76920296060781340324975746860, −18.21324685159535169645627379191, −17.75343423278250220454729782949, −16.92264916008733610121958362645, −16.77693473848908654550406060602, −15.82802806170473633912930233477, −15.56248309308998716630007204974, −14.43163195458328134002338975151, −13.50004648340396989320504396892, −12.33488696765597057065259773085, −11.87534745135013107676801105826, −10.94615612885111038732160876916, −10.79350187367916155153671232394, −9.422303413938473441171880689948, −9.036487932412074199687555304026, −7.55228244623799316615009824381, −7.29041351208685469584643073213, −5.87627034365529106562658369692, −4.38815572194696855867145714346, 4.38815572194696855867145714346, 5.87627034365529106562658369692, 7.29041351208685469584643073213, 7.55228244623799316615009824381, 9.036487932412074199687555304026, 9.422303413938473441171880689948, 10.79350187367916155153671232394, 10.94615612885111038732160876916, 11.87534745135013107676801105826, 12.33488696765597057065259773085, 13.50004648340396989320504396892, 14.43163195458328134002338975151, 15.56248309308998716630007204974, 15.82802806170473633912930233477, 16.77693473848908654550406060602, 16.92264916008733610121958362645, 17.75343423278250220454729782949, 18.21324685159535169645627379191, 18.76920296060781340324975746860, 19.49325330313906922783187166826

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