Properties

Label 4-374e2-1.1-c1e2-0-0
Degree $4$
Conductor $139876$
Sign $-1$
Analytic cond. $8.91861$
Root an. cond. $1.72812$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4-s + 6·9-s + 6·11-s − 4·12-s + 16-s − 10·25-s + 4·27-s − 8·31-s − 24·33-s + 6·36-s − 8·37-s + 6·44-s − 4·48-s + 2·49-s − 12·53-s + 64-s + 16·67-s + 40·75-s − 37·81-s − 12·89-s + 32·93-s + 28·97-s + 36·99-s − 10·100-s − 32·103-s + 4·108-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 2·9-s + 1.80·11-s − 1.15·12-s + 1/4·16-s − 2·25-s + 0.769·27-s − 1.43·31-s − 4.17·33-s + 36-s − 1.31·37-s + 0.904·44-s − 0.577·48-s + 2/7·49-s − 1.64·53-s + 1/8·64-s + 1.95·67-s + 4.61·75-s − 4.11·81-s − 1.27·89-s + 3.31·93-s + 2.84·97-s + 3.61·99-s − 100-s − 3.15·103-s + 0.384·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(139876\)    =    \(2^{2} \cdot 11^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(8.91861\)
Root analytic conductor: \(1.72812\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 139876,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 - 6 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−9.286071988264369821451329617999, −8.624363935884874176037324123450, −8.076455834435977550787516164955, −7.36497483724733271440784650514, −6.72084452362773307312329845344, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −5.67320794510931157328652310045, −5.26894992853838733045784795081, −4.56197836266380508635038039003, −3.90229547122613769308947697962, −3.34611668167428925701071579672, −2.05022911292484155356003327044, −1.25284776655083263545298048613, 0, 1.25284776655083263545298048613, 2.05022911292484155356003327044, 3.34611668167428925701071579672, 3.90229547122613769308947697962, 4.56197836266380508635038039003, 5.26894992853838733045784795081, 5.67320794510931157328652310045, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 6.72084452362773307312329845344, 7.36497483724733271440784650514, 8.076455834435977550787516164955, 8.624363935884874176037324123450, 9.286071988264369821451329617999

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