Properties

Label 4-9771-1.1-c1e2-0-0
Degree $4$
Conductor $9771$
Sign $1$
Analytic cond. $0.623007$
Root an. cond. $0.888430$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s − 4·7-s + 4·9-s + 6·10-s − 3·12-s − 9·13-s + 8·14-s + 9·15-s + 16-s − 17-s − 8·18-s − 2·19-s − 3·20-s + 12·21-s − 6·23-s + 5·25-s + 18·26-s − 4·28-s + 4·29-s − 18·30-s − 2·31-s + 2·32-s + 2·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s − 1.51·7-s + 4/3·9-s + 1.89·10-s − 0.866·12-s − 2.49·13-s + 2.13·14-s + 2.32·15-s + 1/4·16-s − 0.242·17-s − 1.88·18-s − 0.458·19-s − 0.670·20-s + 2.61·21-s − 1.25·23-s + 25-s + 3.53·26-s − 0.755·28-s + 0.742·29-s − 3.28·30-s − 0.359·31-s + 0.353·32-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9771 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9771 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9771\)    =    \(3 \cdot 3257\)
Sign: $1$
Analytic conductor: \(0.623007\)
Root analytic conductor: \(0.888430\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9771,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
3257$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 42 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 49 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T - p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + 60 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 72 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 97 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 42 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 16 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 18 T + 192 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - T + 92 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 18 T + 187 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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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

−17.2590093325, −16.7763123955, −16.2169561575, −16.0198635677, −15.5086970488, −14.8556945919, −14.3687723610, −13.4138077971, −12.7162123620, −12.2019329232, −12.0347879712, −11.6978179686, −10.8132485252, −10.3009718480, −9.87778390942, −9.58727224607, −8.63233969503, −8.27395653010, −7.25877091472, −7.04961990411, −6.37160296631, −5.54192015431, −4.80395821956, −4.02156733105, −2.80946997545, 0, 0, 2.80946997545, 4.02156733105, 4.80395821956, 5.54192015431, 6.37160296631, 7.04961990411, 7.25877091472, 8.27395653010, 8.63233969503, 9.58727224607, 9.87778390942, 10.3009718480, 10.8132485252, 11.6978179686, 12.0347879712, 12.2019329232, 12.7162123620, 13.4138077971, 14.3687723610, 14.8556945919, 15.5086970488, 16.0198635677, 16.2169561575, 16.7763123955, 17.2590093325

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