Properties

Label 4-3584-1.1-c1e2-0-0
Degree $4$
Conductor $3584$
Sign $1$
Analytic cond. $0.228518$
Root an. cond. $0.691401$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 2·9-s − 4·17-s − 6·25-s − 8·31-s + 4·41-s + 8·47-s − 6·49-s − 2·63-s + 24·71-s − 4·73-s + 8·79-s − 5·81-s − 4·89-s + 12·97-s + 4·113-s + 4·119-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 0.377·7-s + 2/3·9-s − 0.970·17-s − 6/5·25-s − 1.43·31-s + 0.624·41-s + 1.16·47-s − 6/7·49-s − 0.251·63-s + 2.84·71-s − 0.468·73-s + 0.900·79-s − 5/9·81-s − 0.423·89-s + 1.21·97-s + 0.376·113-s + 0.366·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.228518\)
Root analytic conductor: \(0.691401\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3584} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7520824033\)
\(L(\frac12)\) \(\approx\) \(0.7520824033\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−12.78633921453067156863824121965, −12.09529846225191096652301766225, −11.36706469231570653953875683954, −10.88295365828671585620615534461, −10.22660277298512962233438113323, −9.500545903444563058216208900345, −9.159683736269665077552023131988, −8.261310410598806618923618858611, −7.54803819142357479327909033553, −6.91428840369939702746087669586, −6.19435508441423655070038710620, −5.36758014632993525220772695623, −4.36293129657109503874157130341, −3.59892807569770409540680161622, −2.14711656048778932954981546455, 2.14711656048778932954981546455, 3.59892807569770409540680161622, 4.36293129657109503874157130341, 5.36758014632993525220772695623, 6.19435508441423655070038710620, 6.91428840369939702746087669586, 7.54803819142357479327909033553, 8.261310410598806618923618858611, 9.159683736269665077552023131988, 9.500545903444563058216208900345, 10.22660277298512962233438113323, 10.88295365828671585620615534461, 11.36706469231570653953875683954, 12.09529846225191096652301766225, 12.78633921453067156863824121965

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