Properties

Label 4-3179-1.1-c1e2-0-0
Degree $4$
Conductor $3179$
Sign $-1$
Analytic cond. $0.202695$
Root an. cond. $0.670982$
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 − 3·3-s + 4-s − 5-s + 3·6-s − 2·7-s − 3·8-s + 3·9-s + 10-s − 3·11-s − 3·12-s + 2·14-s + 3·15-s + 16-s + 6·17-s − 3·18-s − 6·19-s − 20-s + 6·21-s + 3·22-s − 3·23-s + 9·24-s + 3·25-s − 2·28-s − 10·29-s − 3·30-s + 31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s − 0.755·7-s − 1.06·8-s + 9-s + 0.316·10-s − 0.904·11-s − 0.866·12-s + 0.534·14-s + 0.774·15-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.37·19-s − 0.223·20-s + 1.30·21-s + 0.639·22-s − 0.625·23-s + 1.83·24-s + 3/5·25-s − 0.377·28-s − 1.85·29-s − 0.547·30-s + 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3179\)    =    \(11 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(0.202695\)
Root analytic conductor: \(0.670982\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3179,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 5 T + 70 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$D_{4}$ \( 1 + 11 T + 66 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 18 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 138 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T - 26 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 46 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - T - 76 T^{2} - p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 19 T + 246 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
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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

−18.5442796922, −17.7894353050, −17.3369791113, −16.8509015539, −16.4971637941, −16.0567369636, −15.4844073041, −14.9687431862, −14.3521307146, −13.2945260774, −12.7553104472, −12.1869122997, −11.9172420498, −11.1187737809, −10.7809006626, −10.2542041053, −9.43858091797, −8.91196293150, −7.87617027012, −7.40971289994, −6.34288927105, −5.96799060347, −5.42058720580, −4.17111998613, −2.82739846570, 0, 2.82739846570, 4.17111998613, 5.42058720580, 5.96799060347, 6.34288927105, 7.40971289994, 7.87617027012, 8.91196293150, 9.43858091797, 10.2542041053, 10.7809006626, 11.1187737809, 11.9172420498, 12.1869122997, 12.7553104472, 13.2945260774, 14.3521307146, 14.9687431862, 15.4844073041, 16.0567369636, 16.4971637941, 16.8509015539, 17.3369791113, 17.7894353050, 18.5442796922

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