Properties

Label 4-1680e2-1.1-c1e2-0-11
Degree $4$
Conductor $2822400$
Sign $1$
Analytic cond. $179.958$
Root an. cond. $3.66263$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 9-s + 4·11-s − 4·19-s − 25-s + 12·29-s − 12·31-s + 12·41-s + 2·45-s − 49-s − 8·55-s − 16·59-s + 20·61-s + 28·71-s + 8·79-s + 81-s − 20·89-s + 8·95-s − 4·99-s − 20·101-s + 12·109-s − 10·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s − 1/3·9-s + 1.20·11-s − 0.917·19-s − 1/5·25-s + 2.22·29-s − 2.15·31-s + 1.87·41-s + 0.298·45-s − 1/7·49-s − 1.07·55-s − 2.08·59-s + 2.56·61-s + 3.32·71-s + 0.900·79-s + 1/9·81-s − 2.11·89-s + 0.820·95-s − 0.402·99-s − 1.99·101-s + 1.14·109-s − 0.909·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2822400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(179.958\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2822400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.589057804\)
\(L(\frac12)\) \(\approx\) \(1.589057804\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.566878562646041423042272257843, −9.192729664935726728486477390928, −8.706378588763541812595149100941, −8.237814223888623306962877127107, −8.128300178415464168042752227638, −7.68348367633882874211988838190, −6.93501376367607572373007075236, −6.87958100851501840619047971212, −6.50385811052336219938951211551, −5.78161833934511795517188576961, −5.67259094331321193561409582100, −4.94792711066659320369407554227, −4.32603589485085588157140105334, −4.26572485536915366952131009732, −3.57095466389320635486728011979, −3.36994224273942612289903640158, −2.50892694688275645692797230169, −2.10433332557334575626803533222, −1.24768106070456405160249441195, −0.52566280255827036314115637280, 0.52566280255827036314115637280, 1.24768106070456405160249441195, 2.10433332557334575626803533222, 2.50892694688275645692797230169, 3.36994224273942612289903640158, 3.57095466389320635486728011979, 4.26572485536915366952131009732, 4.32603589485085588157140105334, 4.94792711066659320369407554227, 5.67259094331321193561409582100, 5.78161833934511795517188576961, 6.50385811052336219938951211551, 6.87958100851501840619047971212, 6.93501376367607572373007075236, 7.68348367633882874211988838190, 8.128300178415464168042752227638, 8.237814223888623306962877127107, 8.706378588763541812595149100941, 9.192729664935726728486477390928, 9.566878562646041423042272257843

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