Properties

Label 4-3584e2-1.1-c1e2-0-28
Degree $4$
Conductor $12845056$
Sign $1$
Analytic cond. $819.011$
Root an. cond. $5.34961$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 4·9-s + 8·23-s − 10·25-s + 8·31-s − 12·41-s − 8·47-s + 3·49-s + 8·63-s − 16·71-s − 16·73-s − 8·79-s + 7·81-s − 16·89-s − 16·97-s − 8·103-s − 20·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 16·161-s + 163-s + ⋯
L(s)  = 1  − 0.755·7-s − 4/3·9-s + 1.66·23-s − 2·25-s + 1.43·31-s − 1.87·41-s − 1.16·47-s + 3/7·49-s + 1.00·63-s − 1.89·71-s − 1.87·73-s − 0.900·79-s + 7/9·81-s − 1.69·89-s − 1.62·97-s − 0.788·103-s − 1.88·113-s − 1.81·121-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 + 0.0798·157-s − 1.26·161-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12845056\)    =    \(2^{18} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(819.011\)
Root analytic conductor: \(5.34961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 12845056,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 132 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−8.274713950974097415960695304009, −8.190480951580836437613048725913, −7.66991169718480500421481745564, −7.22516206329449014700240201827, −6.70844020939302644511004025432, −6.68742467776977706985105302065, −6.01756545725404539509611003218, −5.83259128796665629065391335799, −5.36637039550000827660304201777, −5.11869883522352165316338792310, −4.40463108894275388966515453718, −4.23559821852181206285700902426, −3.49910126923243332093908676315, −3.21852726623393750657443041195, −2.73840642854949194594715020125, −2.56702188832366835142854182208, −1.63275684334708465729913804750, −1.24145913859199717675211189361, 0, 0, 1.24145913859199717675211189361, 1.63275684334708465729913804750, 2.56702188832366835142854182208, 2.73840642854949194594715020125, 3.21852726623393750657443041195, 3.49910126923243332093908676315, 4.23559821852181206285700902426, 4.40463108894275388966515453718, 5.11869883522352165316338792310, 5.36637039550000827660304201777, 5.83259128796665629065391335799, 6.01756545725404539509611003218, 6.68742467776977706985105302065, 6.70844020939302644511004025432, 7.22516206329449014700240201827, 7.66991169718480500421481745564, 8.190480951580836437613048725913, 8.274713950974097415960695304009

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