Properties

Label 4-1792e2-1.1-c1e2-0-17
Degree $4$
Conductor $3211264$
Sign $1$
Analytic cond. $204.752$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 4·9-s − 12·17-s + 12·23-s − 8·25-s − 8·31-s − 12·41-s + 3·49-s + 8·63-s − 4·73-s + 16·79-s + 7·81-s − 12·89-s − 20·97-s + 8·103-s − 24·113-s + 24·119-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s − 24·161-s + ⋯
L(s)  = 1  − 0.755·7-s − 4/3·9-s − 2.91·17-s + 2.50·23-s − 8/5·25-s − 1.43·31-s − 1.87·41-s + 3/7·49-s + 1.00·63-s − 0.468·73-s + 1.80·79-s + 7/9·81-s − 1.27·89-s − 2.03·97-s + 0.788·103-s − 2.25·113-s + 2.20·119-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.88·153-s + 0.0798·157-s − 1.89·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3211264\)    =    \(2^{16} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(204.752\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3211264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−8.940924759521778181505379343724, −8.917248519513164165984406257004, −8.283228406207846638253734639734, −8.164877637411407172200609504582, −7.28608270441302365439431710489, −7.08915894283415908958162536442, −6.63180254507497902809563536380, −6.46212586418590935164233249435, −5.76597064740174175738918312023, −5.56776808203097827554404418361, −4.83844996195026524900711874308, −4.78778939165836108718564571290, −3.86964436952725826027560565534, −3.67960519229877972775828601242, −3.02895939590336338139094924453, −2.53469516521731775666477050470, −2.18371094795830213388608812376, −1.37582359136721476244828693567, 0, 0, 1.37582359136721476244828693567, 2.18371094795830213388608812376, 2.53469516521731775666477050470, 3.02895939590336338139094924453, 3.67960519229877972775828601242, 3.86964436952725826027560565534, 4.78778939165836108718564571290, 4.83844996195026524900711874308, 5.56776808203097827554404418361, 5.76597064740174175738918312023, 6.46212586418590935164233249435, 6.63180254507497902809563536380, 7.08915894283415908958162536442, 7.28608270441302365439431710489, 8.164877637411407172200609504582, 8.283228406207846638253734639734, 8.917248519513164165984406257004, 8.940924759521778181505379343724

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