Properties

Label 4-4056e2-1.1-c1e2-0-8
Degree $4$
Conductor $16451136$
Sign $1$
Analytic cond. $1048.93$
Root an. cond. $5.69098$
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  + 2·3-s + 3·9-s + 12·17-s − 8·23-s − 6·25-s + 4·27-s − 12·29-s + 8·43-s − 2·49-s + 24·51-s + 12·53-s − 12·61-s − 16·69-s − 12·75-s + 5·81-s − 24·87-s + 28·101-s − 32·103-s + 16·107-s − 20·113-s + 18·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 4·147-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 2.91·17-s − 1.66·23-s − 6/5·25-s + 0.769·27-s − 2.22·29-s + 1.21·43-s − 2/7·49-s + 3.36·51-s + 1.64·53-s − 1.53·61-s − 1.92·69-s − 1.38·75-s + 5/9·81-s − 2.57·87-s + 2.78·101-s − 3.15·103-s + 1.54·107-s − 1.88·113-s + 1.63·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.329·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16451136\)    =    \(2^{6} \cdot 3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1048.93\)
Root analytic conductor: \(5.69098\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16451136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.115473333\)
\(L(\frac12)\) \(\approx\) \(4.115473333\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
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

−8.598521452864360955829862839767, −8.062544245983616492325864985018, −7.80135093654205926373464353434, −7.71717408960355757698576197183, −7.39745331628988825392471511402, −7.02595923762181946990791318183, −6.30424164455667052840375526384, −6.00349448940733312872871203508, −5.54753876115079261667719401602, −5.50894190077442623009113207277, −4.88763169369792174593110492005, −4.04329914482921857822801261330, −4.04211585991692898604170277347, −3.70695839801149713489726659863, −3.06956472516852432533359385506, −2.94033016352601970521626431241, −2.02473872664100069897206355721, −1.92788264923453754657444941782, −1.30503241743468739708169986950, −0.53877028193135650714978210832, 0.53877028193135650714978210832, 1.30503241743468739708169986950, 1.92788264923453754657444941782, 2.02473872664100069897206355721, 2.94033016352601970521626431241, 3.06956472516852432533359385506, 3.70695839801149713489726659863, 4.04211585991692898604170277347, 4.04329914482921857822801261330, 4.88763169369792174593110492005, 5.50894190077442623009113207277, 5.54753876115079261667719401602, 6.00349448940733312872871203508, 6.30424164455667052840375526384, 7.02595923762181946990791318183, 7.39745331628988825392471511402, 7.71717408960355757698576197183, 7.80135093654205926373464353434, 8.062544245983616492325864985018, 8.598521452864360955829862839767

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