Properties

Label 4-10976-1.1-c1e2-0-0
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 − 3·8-s + 2·9-s + 4·11-s + 14-s − 16-s + 2·18-s + 4·22-s + 4·23-s + 6·25-s − 28-s − 16·29-s + 5·32-s − 2·36-s − 8·37-s − 12·43-s − 4·44-s + 4·46-s + 49-s + 6·50-s − 4·53-s − 3·56-s − 16·58-s + 2·63-s + 7·64-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 2/3·9-s + 1.20·11-s + 0.267·14-s − 1/4·16-s + 0.471·18-s + 0.852·22-s + 0.834·23-s + 6/5·25-s − 0.188·28-s − 2.97·29-s + 0.883·32-s − 1/3·36-s − 1.31·37-s − 1.82·43-s − 0.603·44-s + 0.589·46-s + 1/7·49-s + 0.848·50-s − 0.549·53-s − 0.400·56-s − 2.10·58-s + 0.251·63-s + 7/8·64-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

\(L(1)\) \(\approx\) \(1.312347190\)
\(L(\frac12)\) \(\approx\) \(1.312347190\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
7$C_1$ \( 1 - T \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 170 T^{2} + p^{2} T^{4} \)
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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

−11.55774916501801077439355514062, −11.03977963834923531862241725353, −10.34524720863834566815179261268, −9.647618757933168330531102498109, −9.131136714568140745959660783416, −8.806367131463550994402583921281, −8.043363223920110932501204421826, −7.13163420460168122437706237711, −6.80582753422208303503438484081, −5.91296680458355296614584906914, −5.20926022870918845475327495580, −4.65441163247218732104618162078, −3.83458207890807408696497541723, −3.29509910714828509680744318019, −1.66688589248796079219930833540, 1.66688589248796079219930833540, 3.29509910714828509680744318019, 3.83458207890807408696497541723, 4.65441163247218732104618162078, 5.20926022870918845475327495580, 5.91296680458355296614584906914, 6.80582753422208303503438484081, 7.13163420460168122437706237711, 8.043363223920110932501204421826, 8.806367131463550994402583921281, 9.131136714568140745959660783416, 9.647618757933168330531102498109, 10.34524720863834566815179261268, 11.03977963834923531862241725353, 11.55774916501801077439355514062

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