Properties

Label 4-1216e2-1.1-c1e2-0-8
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
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 − 2·9-s + 8·13-s − 6·17-s − 7·25-s + 4·29-s − 20·37-s + 20·41-s − 4·45-s − 5·49-s + 8·53-s + 26·61-s + 16·65-s + 18·73-s − 5·81-s − 12·85-s + 24·89-s − 16·97-s + 20·101-s − 20·113-s − 16·117-s + 3·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s + 2.21·13-s − 1.45·17-s − 7/5·25-s + 0.742·29-s − 3.28·37-s + 3.12·41-s − 0.596·45-s − 5/7·49-s + 1.09·53-s + 3.32·61-s + 1.98·65-s + 2.10·73-s − 5/9·81-s − 1.30·85-s + 2.54·89-s − 1.62·97-s + 1.99·101-s − 1.88·113-s − 1.47·117-s + 3/11·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1478656} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.415391886\)
\(L(\frac12)\) \(\approx\) \(2.415391886\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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

−8.121865683855346250917871835791, −7.46802549489449929304458830547, −6.89999375237673531400272497166, −6.39486340421971046323119110374, −6.35171638937504994192666881666, −5.61135260065455155907566257543, −5.55162300458149510222680140561, −4.93085486739421392203871722376, −4.17947794613570531538770424508, −3.73571330395388729025783799541, −3.51647821375355615052320307776, −2.48496537170117614857619865014, −2.22354879090908877160738042469, −1.55214010532178794455093749689, −0.67525660115678139253233977173, 0.67525660115678139253233977173, 1.55214010532178794455093749689, 2.22354879090908877160738042469, 2.48496537170117614857619865014, 3.51647821375355615052320307776, 3.73571330395388729025783799541, 4.17947794613570531538770424508, 4.93085486739421392203871722376, 5.55162300458149510222680140561, 5.61135260065455155907566257543, 6.35171638937504994192666881666, 6.39486340421971046323119110374, 6.89999375237673531400272497166, 7.46802549489449929304458830547, 8.121865683855346250917871835791

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