Properties

Label 4-1078e2-1.1-c1e2-0-41
Degree $4$
Conductor $1162084$
Sign $-1$
Analytic cond. $74.0954$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 2·9-s − 3·11-s + 5·16-s + 4·18-s − 6·22-s − 6·23-s − 8·25-s + 6·32-s + 6·36-s − 5·37-s − 9·43-s − 9·44-s − 12·46-s − 16·50-s − 9·53-s + 7·64-s − 7·67-s + 3·71-s + 8·72-s − 10·74-s − 24·79-s − 5·81-s − 18·86-s − 12·88-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2/3·9-s − 0.904·11-s + 5/4·16-s + 0.942·18-s − 1.27·22-s − 1.25·23-s − 8/5·25-s + 1.06·32-s + 36-s − 0.821·37-s − 1.37·43-s − 1.35·44-s − 1.76·46-s − 2.26·50-s − 1.23·53-s + 7/8·64-s − 0.855·67-s + 0.356·71-s + 0.942·72-s − 1.16·74-s − 2.70·79-s − 5/9·81-s − 1.94·86-s − 1.27·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1162084\)    =    \(2^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(74.0954\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \)
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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

−7.66109140875782292797439734506, −7.38022491709403490834438782513, −6.82325337565643340745723175151, −6.39702282875957859237193755899, −5.98600293927349867048524374338, −5.44179789280298378384022589030, −5.25712198345805760925672550290, −4.53946070710420453213655625238, −4.20599149051304473265306609294, −3.79497807382036326947700718378, −3.14617807019897249368473777505, −2.72126381842469168923278952382, −1.85505636399294786038277846477, −1.64121900689994963251181644061, 0, 1.64121900689994963251181644061, 1.85505636399294786038277846477, 2.72126381842469168923278952382, 3.14617807019897249368473777505, 3.79497807382036326947700718378, 4.20599149051304473265306609294, 4.53946070710420453213655625238, 5.25712198345805760925672550290, 5.44179789280298378384022589030, 5.98600293927349867048524374338, 6.39702282875957859237193755899, 6.82325337565643340745723175151, 7.38022491709403490834438782513, 7.66109140875782292797439734506

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