Properties

Label 4-48e4-1.1-c3e2-0-29
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $18479.7$
Root an. cond. $11.6593$
Motivic weight $3$
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  − 40·11-s + 68·17-s + 104·19-s − 242·25-s + 52·41-s + 504·43-s − 486·49-s − 728·59-s + 1.25e3·67-s + 676·73-s − 2.07e3·83-s − 468·89-s − 356·97-s − 2.80e3·107-s − 2.75e3·113-s − 1.46e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.82e3·169-s + 173-s + ⋯
L(s)  = 1  − 1.09·11-s + 0.970·17-s + 1.25·19-s − 1.93·25-s + 0.198·41-s + 1.78·43-s − 1.41·49-s − 1.60·59-s + 2.29·67-s + 1.08·73-s − 2.74·83-s − 0.557·89-s − 0.372·97-s − 2.53·107-s − 2.29·113-s − 1.09·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.28·169-s + 0.000439·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(18479.7\)
Root analytic conductor: \(11.6593\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5308416,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 242 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 486 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 20 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 2826 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 52 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 20462 T^{2} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 8450 T^{2} + p^{6} T^{4} \)
31$C_2^2$ \( 1 + 47414 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 27578 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 26 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 252 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 88574 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 166894 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 364 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 - 86838 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 628 T + p^{3} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 604430 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 - 338 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 363350 T^{2} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 1036 T + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 + 234 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 178 T + p^{3} T^{2} )^{2} \)
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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

−8.125709043154121783727582121406, −8.090216327306470616308953274082, −7.68474971848931030322070945816, −7.60214241522955514365271994062, −6.82686261435344295353264715231, −6.65903911995254774118057232156, −5.95647068821454249340170427658, −5.65579182131629756627970926650, −5.30470407747296080935945734506, −5.12345080342747537723405474849, −4.28611840630863406825542033921, −4.08756416781764431283351279878, −3.47710281783325012430758249252, −3.08546346978539821710780490055, −2.54538652406231311701105519025, −2.20279298025413948886632278745, −1.29573481456013158198339857232, −1.15254562580518592920482503340, 0, 0, 1.15254562580518592920482503340, 1.29573481456013158198339857232, 2.20279298025413948886632278745, 2.54538652406231311701105519025, 3.08546346978539821710780490055, 3.47710281783325012430758249252, 4.08756416781764431283351279878, 4.28611840630863406825542033921, 5.12345080342747537723405474849, 5.30470407747296080935945734506, 5.65579182131629756627970926650, 5.95647068821454249340170427658, 6.65903911995254774118057232156, 6.82686261435344295353264715231, 7.60214241522955514365271994062, 7.68474971848931030322070945816, 8.090216327306470616308953274082, 8.125709043154121783727582121406

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