Properties

Label 4-160e2-1.1-c3e2-0-7
Degree $4$
Conductor $25600$
Sign $1$
Analytic cond. $89.1193$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·5-s − 34·9-s − 124·13-s − 92·17-s + 75·25-s − 180·29-s − 428·37-s − 20·41-s + 340·45-s + 294·49-s − 1.35e3·53-s + 500·61-s + 1.24e3·65-s + 1.04e3·73-s + 427·81-s + 920·85-s + 1.94e3·89-s − 1.86e3·97-s − 1.20e3·101-s + 4.30e3·109-s − 4.36e3·113-s + 4.21e3·117-s − 2.58e3·121-s − 500·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.25·9-s − 2.64·13-s − 1.31·17-s + 3/5·25-s − 1.15·29-s − 1.90·37-s − 0.0761·41-s + 1.12·45-s + 6/7·49-s − 3.51·53-s + 1.04·61-s + 2.36·65-s + 1.67·73-s + 0.585·81-s + 1.17·85-s + 2.31·89-s − 1.95·97-s − 1.18·101-s + 3.78·109-s − 3.63·113-s + 3.33·117-s − 1.93·121-s − 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25600\)    =    \(2^{10} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(89.1193\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{160} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 25600,\ (\ :3/2, 3/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + p T )^{2} \)
good3$C_2^2$ \( 1 + 34 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 6 p^{2} T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 2582 T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 62 T + p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 46 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( 1 + 2198 T^{2} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 12646 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 90 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 36462 T^{2} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 214 T + p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 10 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 154514 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 49226 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 678 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 241478 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 250 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 599106 T^{2} + p^{6} T^{4} \)
71$C_2^2$ \( 1 + 581342 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 - 522 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 217758 T^{2} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 999074 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 970 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 934 T + p^{3} T^{2} )^{2} \)
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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

−12.14417407189950667783235657394, −11.72635719987079053075994375186, −11.19990040512809189244998901798, −10.85092715792755929576333917388, −10.18354381072050069418901439625, −9.506101788907985375513364111873, −9.120904275493744894244518951720, −8.590225515539206919452026639226, −7.80295355920318112079605761779, −7.64245355927736891419684440360, −6.82808572228664515094071631921, −6.47073839427840834831047605747, −5.25780113204589902706074526859, −5.14506212575153012544161861903, −4.34505219403782920223278302616, −3.50053540682691292918108792216, −2.71009757244035338446943905309, −2.05225850346205488929694359608, 0, 0, 2.05225850346205488929694359608, 2.71009757244035338446943905309, 3.50053540682691292918108792216, 4.34505219403782920223278302616, 5.14506212575153012544161861903, 5.25780113204589902706074526859, 6.47073839427840834831047605747, 6.82808572228664515094071631921, 7.64245355927736891419684440360, 7.80295355920318112079605761779, 8.590225515539206919452026639226, 9.120904275493744894244518951720, 9.506101788907985375513364111873, 10.18354381072050069418901439625, 10.85092715792755929576333917388, 11.19990040512809189244998901798, 11.72635719987079053075994375186, 12.14417407189950667783235657394

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