Properties

Label 4-1059968-1.1-c1e2-0-2
Degree $4$
Conductor $1059968$
Sign $1$
Analytic cond. $67.5844$
Root an. cond. $2.86722$
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-s + 2·3-s + 4-s + 2·6-s + 8-s − 3·9-s − 6·11-s + 2·12-s + 16-s − 3·18-s + 4·19-s − 6·22-s + 2·24-s − 10·25-s − 14·27-s + 32-s − 12·33-s − 3·36-s + 4·38-s + 6·41-s + 16·43-s − 6·44-s + 2·48-s + 49-s − 10·50-s − 14·54-s + 8·57-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s − 9-s − 1.80·11-s + 0.577·12-s + 1/4·16-s − 0.707·18-s + 0.917·19-s − 1.27·22-s + 0.408·24-s − 2·25-s − 2.69·27-s + 0.176·32-s − 2.08·33-s − 1/2·36-s + 0.648·38-s + 0.937·41-s + 2.43·43-s − 0.904·44-s + 0.288·48-s + 1/7·49-s − 1.41·50-s − 1.90·54-s + 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1059968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1059968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1059968\)    =    \(2^{7} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(67.5844\)
Root analytic conductor: \(2.86722\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1059968,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.263772781\)
\(L(\frac12)\) \(\approx\) \(3.263772781\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−7.974204239601213188956236456444, −7.63380170955425435565378568582, −7.60007770624079585422104671113, −6.81334322707606658549650565225, −6.10743185313712551583625613621, −5.71034209994038661023746867842, −5.49329025649653204207581393927, −5.06509234546123458101387314677, −4.34218533360235296422646954393, −3.60858467418748472558474969025, −3.54211448492338401696398190009, −2.72564141720431011383752495662, −2.33000973337707432224237511883, −2.17542049731475021336825361662, −0.63890669930845811510933037152, 0.63890669930845811510933037152, 2.17542049731475021336825361662, 2.33000973337707432224237511883, 2.72564141720431011383752495662, 3.54211448492338401696398190009, 3.60858467418748472558474969025, 4.34218533360235296422646954393, 5.06509234546123458101387314677, 5.49329025649653204207581393927, 5.71034209994038661023746867842, 6.10743185313712551583625613621, 6.81334322707606658549650565225, 7.60007770624079585422104671113, 7.63380170955425435565378568582, 7.974204239601213188956236456444

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