Properties

Label 4-80e4-1.1-c1e2-0-17
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
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  + 2·3-s + 3·9-s + 6·11-s + 6·17-s + 2·19-s + 10·27-s + 12·33-s − 6·41-s + 20·43-s − 14·49-s + 12·51-s + 4·57-s − 12·59-s − 14·67-s + 2·73-s + 20·81-s + 18·83-s + 18·89-s + 20·97-s + 18·99-s + 6·107-s − 18·113-s + 11·121-s − 12·123-s + 127-s + 40·129-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 1.80·11-s + 1.45·17-s + 0.458·19-s + 1.92·27-s + 2.08·33-s − 0.937·41-s + 3.04·43-s − 2·49-s + 1.68·51-s + 0.529·57-s − 1.56·59-s − 1.71·67-s + 0.234·73-s + 20/9·81-s + 1.97·83-s + 1.90·89-s + 2.03·97-s + 1.80·99-s + 0.580·107-s − 1.69·113-s + 121-s − 1.08·123-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.613992160\)
\(L(\frac12)\) \(\approx\) \(7.613992160\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.089072140521162259181364847543, −7.76901248556106673302845362710, −7.54096342656900961729221781560, −7.45724747765445722699510538208, −6.66012653744537374402702732222, −6.46944451060232970931396094354, −6.31542161500589761777841979758, −5.78183608835195473741978321896, −5.32066745230054702692236164958, −4.77010727617649397326285679763, −4.65628553618816095860144623806, −4.05242372612926296745183654997, −3.73149065917015912549951902318, −3.39705371486053473809764217363, −3.04509386713973164489791517976, −2.68052534004676599074607342335, −2.03352257076304928672805934639, −1.55296673083587789218299526210, −1.17347853237472412003519448556, −0.70387167378117777364371662144, 0.70387167378117777364371662144, 1.17347853237472412003519448556, 1.55296673083587789218299526210, 2.03352257076304928672805934639, 2.68052534004676599074607342335, 3.04509386713973164489791517976, 3.39705371486053473809764217363, 3.73149065917015912549951902318, 4.05242372612926296745183654997, 4.65628553618816095860144623806, 4.77010727617649397326285679763, 5.32066745230054702692236164958, 5.78183608835195473741978321896, 6.31542161500589761777841979758, 6.46944451060232970931396094354, 6.66012653744537374402702732222, 7.45724747765445722699510538208, 7.54096342656900961729221781560, 7.76901248556106673302845362710, 8.089072140521162259181364847543

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