Properties

Label 4-7889-1.1-c1e2-0-1
Degree $4$
Conductor $7889$
Sign $-1$
Analytic cond. $0.503009$
Root an. cond. $0.842158$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 5-s + 2·6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 13-s − 14-s + 2·15-s − 16-s − 5·17-s − 18-s − 3·19-s − 2·21-s − 22-s − 4·23-s + 2·24-s + 3·25-s + 26-s − 2·27-s − 5·29-s − 2·30-s − 2·31-s + 6·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 0.447·5-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.277·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.688·19-s − 0.436·21-s − 0.213·22-s − 0.834·23-s + 0.408·24-s + 3/5·25-s + 0.196·26-s − 0.384·27-s − 0.928·29-s − 0.365·30-s − 0.359·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7889\)    =    \(7^{3} \cdot 23\)
Sign: $-1$
Analytic conductor: \(0.503009\)
Root analytic conductor: \(0.842158\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 7889,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$ \( 1 - T \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 3 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + 64 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 11 T + 110 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 5 T - 50 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−17.1744804238, −16.9432379204, −16.2872081955, −15.7311846847, −15.2817024718, −14.8288010428, −14.2335185430, −13.4728873380, −13.0351303273, −12.3486333474, −11.7974791568, −11.4241615559, −11.0635906465, −10.4360372403, −9.79867246129, −9.09292700997, −8.70355237272, −8.02278148216, −7.32614173093, −6.48818817932, −6.19254774509, −5.18137785062, −4.59160048458, −3.68556081933, −2.13997380920, 0, 2.13997380920, 3.68556081933, 4.59160048458, 5.18137785062, 6.19254774509, 6.48818817932, 7.32614173093, 8.02278148216, 8.70355237272, 9.09292700997, 9.79867246129, 10.4360372403, 11.0635906465, 11.4241615559, 11.7974791568, 12.3486333474, 13.0351303273, 13.4728873380, 14.2335185430, 14.8288010428, 15.2817024718, 15.7311846847, 16.2872081955, 16.9432379204, 17.1744804238

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