Properties

Label 4-2e12-1.1-c1e2-0-4
Degree $4$
Conductor $4096$
Sign $1$
Analytic cond. $0.261164$
Root an. cond. $0.714872$
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  + 2·3-s − 2·5-s + 2·9-s − 2·11-s − 2·13-s − 4·15-s − 4·17-s − 6·19-s + 2·25-s + 6·27-s + 6·29-s + 16·31-s − 4·33-s + 6·37-s − 4·39-s − 10·43-s − 4·45-s − 16·47-s + 10·49-s − 8·51-s − 10·53-s + 4·55-s − 12·57-s + 6·59-s − 18·61-s + 4·65-s + 10·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 2/3·9-s − 0.603·11-s − 0.554·13-s − 1.03·15-s − 0.970·17-s − 1.37·19-s + 2/5·25-s + 1.15·27-s + 1.11·29-s + 2.87·31-s − 0.696·33-s + 0.986·37-s − 0.640·39-s − 1.52·43-s − 0.596·45-s − 2.33·47-s + 10/7·49-s − 1.12·51-s − 1.37·53-s + 0.539·55-s − 1.58·57-s + 0.781·59-s − 2.30·61-s + 0.496·65-s + 1.22·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $1$
Analytic conductor: \(0.261164\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{64} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4096,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9253180416\)
\(L(\frac12)\) \(\approx\) \(0.9253180416\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−15.15806687836610164216618137463, −14.80890398541489430227393698201, −14.08609491620803761202087545846, −13.71646814124117271064522680575, −12.95059353984951873190801474811, −12.68493327599984495986802133024, −11.79680949406565484562975769216, −11.48778254946706635368882741265, −10.38797986358525437670973004875, −10.29444745064215992476530998944, −9.337445096027797491465193578904, −8.623372325765915389171432951705, −8.094575571028396639247074109329, −7.943227555938203405118009237953, −6.71055742683799316053347084224, −6.44149975752932406076517563358, −4.66177529723384040571861019470, −4.56124503667803827231168171669, −3.19980240997241295571649335002, −2.46719287874121892924646954170, 2.46719287874121892924646954170, 3.19980240997241295571649335002, 4.56124503667803827231168171669, 4.66177529723384040571861019470, 6.44149975752932406076517563358, 6.71055742683799316053347084224, 7.943227555938203405118009237953, 8.094575571028396639247074109329, 8.623372325765915389171432951705, 9.337445096027797491465193578904, 10.29444745064215992476530998944, 10.38797986358525437670973004875, 11.48778254946706635368882741265, 11.79680949406565484562975769216, 12.68493327599984495986802133024, 12.95059353984951873190801474811, 13.71646814124117271064522680575, 14.08609491620803761202087545846, 14.80890398541489430227393698201, 15.15806687836610164216618137463

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