Properties

Label 4-10976-1.1-c1e2-0-2
Degree $4$
Conductor $10976$
Sign $1$
Analytic cond. $0.699839$
Root an. cond. $0.914638$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 2·9-s − 14-s + 16-s + 2·18-s − 8·23-s − 6·25-s − 28-s + 4·29-s + 32-s + 2·36-s + 4·37-s − 8·46-s + 49-s − 6·50-s − 4·53-s − 56-s + 4·58-s − 2·63-s + 64-s − 24·67-s + 16·71-s + 2·72-s + 4·74-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 2/3·9-s − 0.267·14-s + 1/4·16-s + 0.471·18-s − 1.66·23-s − 6/5·25-s − 0.188·28-s + 0.742·29-s + 0.176·32-s + 1/3·36-s + 0.657·37-s − 1.17·46-s + 1/7·49-s − 0.848·50-s − 0.549·53-s − 0.133·56-s + 0.525·58-s − 0.251·63-s + 1/8·64-s − 2.93·67-s + 1.89·71-s + 0.235·72-s + 0.464·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10976\)    =    \(2^{5} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(0.699839\)
Root analytic conductor: \(0.914638\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10976} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.479234028\)
\(L(\frac12)\) \(\approx\) \(1.479234028\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
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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

−11.67243765067819893667519287194, −10.86030205769484123590609978555, −10.38154541301378339571912207256, −9.780964096385823605224897000939, −9.440497965278766401080888076377, −8.466149797686215063234877480194, −7.87080399329017938577900963261, −7.37229758267953838614619646206, −6.54361952531080618560362273781, −6.09461834558935249753190776977, −5.41783473797801293529811804623, −4.41783868169106956331622880408, −4.00514677824647069794495389872, −3.00486518899045402623792404204, −1.88056630179972506147475318583, 1.88056630179972506147475318583, 3.00486518899045402623792404204, 4.00514677824647069794495389872, 4.41783868169106956331622880408, 5.41783473797801293529811804623, 6.09461834558935249753190776977, 6.54361952531080618560362273781, 7.37229758267953838614619646206, 7.87080399329017938577900963261, 8.466149797686215063234877480194, 9.440497965278766401080888076377, 9.780964096385823605224897000939, 10.38154541301378339571912207256, 10.86030205769484123590609978555, 11.67243765067819893667519287194

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