Properties

Label 4-40e4-1.1-c1e2-0-24
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 6·9-s − 4·27-s − 12·41-s + 20·43-s + 14·49-s + 28·67-s − 37·81-s + 36·83-s − 36·89-s + 12·107-s − 14·121-s − 48·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s + 56·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s − 0.769·27-s − 1.87·41-s + 3.04·43-s + 2·49-s + 3.42·67-s − 4.11·81-s + 3.95·83-s − 3.81·89-s + 1.16·107-s − 1.27·121-s − 4.32·123-s + 0.0887·127-s + 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.61·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.075994773\)
\(L(\frac12)\) \(\approx\) \(5.075994773\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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

−9.534829036418252700257629357443, −9.043286683129683219459596029754, −8.730054488693157569360021493151, −8.591579832571673505528003084598, −8.019838620753455863560859146050, −7.82336287633584810161167813566, −7.38715341089691132133790223606, −7.02556548992560164162144608904, −6.49255958603362394750607504820, −5.96858853173716671788493034377, −5.41852473123892389915316951896, −5.14925602938534690765238908263, −4.29569294913590630432367766231, −3.89753377453048086960262528029, −3.64768968698289207823728794986, −3.08735359786260095590437152702, −2.43016051313926583327314117215, −2.42481699904604462119437761230, −1.71493873068809262695085591796, −0.74244896642002666158509946344, 0.74244896642002666158509946344, 1.71493873068809262695085591796, 2.42481699904604462119437761230, 2.43016051313926583327314117215, 3.08735359786260095590437152702, 3.64768968698289207823728794986, 3.89753377453048086960262528029, 4.29569294913590630432367766231, 5.14925602938534690765238908263, 5.41852473123892389915316951896, 5.96858853173716671788493034377, 6.49255958603362394750607504820, 7.02556548992560164162144608904, 7.38715341089691132133790223606, 7.82336287633584810161167813566, 8.019838620753455863560859146050, 8.591579832571673505528003084598, 8.730054488693157569360021493151, 9.043286683129683219459596029754, 9.534829036418252700257629357443

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