Properties

Label 4-103e2-1.1-c1e2-0-0
Degree $4$
Conductor $10609$
Sign $-1$
Analytic cond. $0.676439$
Root an. cond. $0.906895$
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 − 3-s + 5-s + 6-s − 2·7-s − 8-s − 2·9-s − 10-s − 3·13-s + 2·14-s − 15-s − 16-s − 4·17-s + 2·18-s − 3·19-s + 2·21-s − 2·23-s + 24-s + 4·25-s + 3·26-s + 2·27-s + 29-s + 30-s − 2·31-s + 6·32-s + 4·34-s − 2·35-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.832·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.471·18-s − 0.688·19-s + 0.436·21-s − 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.588·26-s + 0.384·27-s + 0.185·29-s + 0.182·30-s − 0.359·31-s + 1.06·32-s + 0.685·34-s − 0.338·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10609\)    =    \(103^{2}\)
Sign: $-1$
Analytic conductor: \(0.676439\)
Root analytic conductor: \(0.906895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 10609,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad103$C_2$ \( 1 - 11 T + p T^{2} \)
good2$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T - 11 T^{2} - p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + 79 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 139 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T - 87 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 128 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 146 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 176 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−16.9780912318, −16.3458410408, −16.0249522816, −15.3073103733, −14.8772803296, −14.3769973081, −13.7690458079, −13.1071794766, −12.8742738906, −12.1447717586, −11.6453013190, −11.1683675382, −10.5754572613, −9.94364033961, −9.57274679628, −8.89658520426, −8.63042346114, −7.79693145314, −6.96805648078, −6.32785257914, −6.06033196706, −5.06737678461, −4.42963356444, −3.14468474273, −2.28158656589, 0, 2.28158656589, 3.14468474273, 4.42963356444, 5.06737678461, 6.06033196706, 6.32785257914, 6.96805648078, 7.79693145314, 8.63042346114, 8.89658520426, 9.57274679628, 9.94364033961, 10.5754572613, 11.1683675382, 11.6453013190, 12.1447717586, 12.8742738906, 13.1071794766, 13.7690458079, 14.3769973081, 14.8772803296, 15.3073103733, 16.0249522816, 16.3458410408, 16.9780912318

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