Properties

Label 4-10976-1.1-c1e2-0-3
Degree $4$
Conductor $10976$
Sign $1$
Analytic cond. $0.699839$
Root an. cond. $0.914638$
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 + 4·3-s + 4-s − 4·6-s − 7-s − 8-s + 6·9-s + 4·12-s + 14-s + 16-s − 6·18-s − 4·19-s − 4·21-s − 4·24-s − 10·25-s − 4·27-s − 28-s − 12·29-s + 8·31-s − 32-s + 6·36-s + 4·37-s + 4·38-s + 4·42-s + 24·47-s + 4·48-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s + 2.30·3-s + 1/2·4-s − 1.63·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 1.15·12-s + 0.267·14-s + 1/4·16-s − 1.41·18-s − 0.917·19-s − 0.872·21-s − 0.816·24-s − 2·25-s − 0.769·27-s − 0.188·28-s − 2.22·29-s + 1.43·31-s − 0.176·32-s + 36-s + 0.657·37-s + 0.648·38-s + 0.617·42-s + 3.50·47-s + 0.577·48-s + 1/7·49-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.328111995\)
\(L(\frac12)\) \(\approx\) \(1.328111995\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−11.59222775413431567626281153330, −10.57696390398788315518055643332, −10.11465006559287644470603673937, −9.392494157732895222790142944171, −9.205467104844532229657218112988, −8.624667079434487291694904422005, −8.182643116839158569055135937831, −7.55563465371540871894331307120, −7.29668061464756137953129595214, −6.12986038892916091786195943044, −5.62168100149868099368912779374, −3.87985229263682105648190826671, −3.82418745638366191622617161116, −2.48514181668809031419660323366, −2.25140513775369021989127014661, 2.25140513775369021989127014661, 2.48514181668809031419660323366, 3.82418745638366191622617161116, 3.87985229263682105648190826671, 5.62168100149868099368912779374, 6.12986038892916091786195943044, 7.29668061464756137953129595214, 7.55563465371540871894331307120, 8.182643116839158569055135937831, 8.624667079434487291694904422005, 9.205467104844532229657218112988, 9.392494157732895222790142944171, 10.11465006559287644470603673937, 10.57696390398788315518055643332, 11.59222775413431567626281153330

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