Properties

Label 4-970-1.1-c1e2-0-0
Degree $4$
Conductor $970$
Sign $1$
Analytic cond. $0.0618480$
Root an. cond. $0.498690$
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 − 3·5-s + 7-s + 8-s + 3·10-s − 5·11-s − 3·13-s − 14-s − 3·16-s − 3·20-s + 5·22-s + 9·23-s + 8·25-s + 3·26-s + 28-s − 4·29-s − 31-s + 5·32-s − 3·35-s + 9·37-s − 3·40-s − 4·41-s + 2·43-s − 5·44-s − 9·46-s − 4·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s − 1.50·11-s − 0.832·13-s − 0.267·14-s − 3/4·16-s − 0.670·20-s + 1.06·22-s + 1.87·23-s + 8/5·25-s + 0.588·26-s + 0.188·28-s − 0.742·29-s − 0.179·31-s + 0.883·32-s − 0.507·35-s + 1.47·37-s − 0.474·40-s − 0.624·41-s + 0.304·43-s − 0.753·44-s − 1.32·46-s − 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 970 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 970 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(970\)    =    \(2 \cdot 5 \cdot 97\)
Sign: $1$
Analytic conductor: \(0.0618480\)
Root analytic conductor: \(0.498690\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 970,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3475154474\)
\(L(\frac12)\) \(\approx\) \(0.3475154474\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \)
29$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$D_{4}$ \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T - 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 6 T - 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$D_{4}$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−19.4386359096, −18.8929149382, −18.5350402938, −17.9690464567, −17.1047375426, −16.7021264024, −16.1605246924, −15.4996253688, −15.0713542355, −14.5976207221, −13.4751902852, −12.9524302057, −12.2948266056, −11.4244789416, −11.0068365658, −10.5128789417, −9.57508637702, −8.72403929332, −7.88618166836, −7.55968001574, −6.86259334572, −5.23325102631, −4.49327541213, −2.85695421779, 2.85695421779, 4.49327541213, 5.23325102631, 6.86259334572, 7.55968001574, 7.88618166836, 8.72403929332, 9.57508637702, 10.5128789417, 11.0068365658, 11.4244789416, 12.2948266056, 12.9524302057, 13.4751902852, 14.5976207221, 15.0713542355, 15.4996253688, 16.1605246924, 16.7021264024, 17.1047375426, 17.9690464567, 18.5350402938, 18.8929149382, 19.4386359096

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