Properties

Label 4-176e2-1.1-c1e2-0-0
Degree $4$
Conductor $30976$
Sign $1$
Analytic cond. $1.97505$
Root an. cond. $1.18548$
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  + 4·7-s − 5·9-s + 12·17-s − 6·23-s − 25-s + 10·31-s − 2·49-s − 20·63-s + 30·71-s − 8·73-s + 4·79-s + 16·81-s − 18·89-s − 14·97-s + 16·103-s − 30·113-s + 48·119-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 60·153-s + 157-s − 24·161-s + ⋯
L(s)  = 1  + 1.51·7-s − 5/3·9-s + 2.91·17-s − 1.25·23-s − 1/5·25-s + 1.79·31-s − 2/7·49-s − 2.51·63-s + 3.56·71-s − 0.936·73-s + 0.450·79-s + 16/9·81-s − 1.90·89-s − 1.42·97-s + 1.57·103-s − 2.82·113-s + 4.40·119-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 4.85·153-s + 0.0798·157-s − 1.89·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30976\)    =    \(2^{8} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1.97505\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 30976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.373460837\)
\(L(\frac12)\) \(\approx\) \(1.373460837\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−10.54980513640919233558709119533, −9.813753565180213813561671136388, −9.700092895780422192241801282926, −8.674866558231347995249063814171, −8.132665807481574488057539990847, −8.050490063857036795242146503456, −7.60852684015633966980596452192, −6.51356460177744683595774086764, −5.96029809065759357817228429363, −5.29396314589658220886171477329, −5.11227132195889762323685903220, −4.04654564288201609443200871032, −3.27381985666804460223319703873, −2.49141172081529131249029798363, −1.28296422834420783986833535835, 1.28296422834420783986833535835, 2.49141172081529131249029798363, 3.27381985666804460223319703873, 4.04654564288201609443200871032, 5.11227132195889762323685903220, 5.29396314589658220886171477329, 5.96029809065759357817228429363, 6.51356460177744683595774086764, 7.60852684015633966980596452192, 8.050490063857036795242146503456, 8.132665807481574488057539990847, 8.674866558231347995249063814171, 9.700092895780422192241801282926, 9.813753565180213813561671136388, 10.54980513640919233558709119533

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