Properties

Label 4-336e2-1.1-c1e2-0-68
Degree $4$
Conductor $112896$
Sign $1$
Analytic cond. $7.19834$
Root an. cond. $1.63797$
Motivic weight $1$
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  − 3-s − 6·5-s + 7-s − 6·11-s + 6·15-s − 12·17-s − 7·19-s − 21-s + 19·25-s + 27-s − 5·31-s + 6·33-s − 6·35-s − 37-s + 6·47-s − 6·49-s + 12·51-s + 36·55-s + 7·57-s + 3·67-s + 15·73-s − 19·75-s − 6·77-s − 27·79-s − 81-s − 12·83-s + 72·85-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.68·5-s + 0.377·7-s − 1.80·11-s + 1.54·15-s − 2.91·17-s − 1.60·19-s − 0.218·21-s + 19/5·25-s + 0.192·27-s − 0.898·31-s + 1.04·33-s − 1.01·35-s − 0.164·37-s + 0.875·47-s − 6/7·49-s + 1.68·51-s + 4.85·55-s + 0.927·57-s + 0.366·67-s + 1.75·73-s − 2.19·75-s − 0.683·77-s − 3.03·79-s − 1/9·81-s − 1.31·83-s + 7.80·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.19834\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 112896,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 27 T + 322 T^{2} + 27 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−11.17658038107463008880336453664, −11.14326503702686611059126841332, −10.64921731435778230999586349950, −10.33315490593684826385948079765, −9.304044307619063852762340233950, −8.717449121805435780937217096209, −8.288780824004539839113294285056, −8.214432093735022090255064633657, −7.46315815379619051630993934752, −7.13540886844588892096243509000, −6.65854050248894508109130516911, −5.97477288614643466619648720596, −5.07860903041432734059681352532, −4.72203094873673997377963322849, −4.16317481024524719337693887434, −3.88913670573558239821495281904, −2.83855154370978815589437294842, −2.18035028887262902051783966225, 0, 0, 2.18035028887262902051783966225, 2.83855154370978815589437294842, 3.88913670573558239821495281904, 4.16317481024524719337693887434, 4.72203094873673997377963322849, 5.07860903041432734059681352532, 5.97477288614643466619648720596, 6.65854050248894508109130516911, 7.13540886844588892096243509000, 7.46315815379619051630993934752, 8.214432093735022090255064633657, 8.288780824004539839113294285056, 8.717449121805435780937217096209, 9.304044307619063852762340233950, 10.33315490593684826385948079765, 10.64921731435778230999586349950, 11.14326503702686611059126841332, 11.17658038107463008880336453664

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