Properties

Label 4-909-1.1-c1e2-0-0
Degree $4$
Conductor $909$
Sign $1$
Analytic cond. $0.0579586$
Root an. cond. $0.490658$
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 − 5-s − 4·7-s − 3·8-s + 9-s + 10-s + 2·11-s + 3·13-s + 4·14-s + 16-s − 17-s − 18-s − 19-s − 20-s − 2·22-s − 23-s + 25-s − 3·26-s − 4·28-s + 4·29-s − 31-s + 32-s + 34-s + 4·35-s + 36-s + 38-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.832·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.426·22-s − 0.208·23-s + 1/5·25-s − 0.588·26-s − 0.755·28-s + 0.742·29-s − 0.179·31-s + 0.176·32-s + 0.171·34-s + 0.676·35-s + 1/6·36-s + 0.162·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 909 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 909 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(909\)    =    \(3^{2} \cdot 101\)
Sign: $1$
Analytic conductor: \(0.0579586\)
Root analytic conductor: \(0.490658\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 909,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3407116942\)
\(L(\frac12)\) \(\approx\) \(0.3407116942\)
\(L(\frac{3}{2})\) 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_1$ \( ( 1 - T )( 1 + T ) \)
101$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$D_{4}$ \( 1 - 3 T - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$D_{4}$ \( 1 + T - 34 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$D_{4}$ \( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 46 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 5 T + 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$D_{4}$ \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 186 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−19.5268623180, −19.0223801228, −18.5939242709, −18.0317193227, −17.3762435820, −16.7376346664, −16.0517876805, −15.7654481631, −15.3332748645, −14.4322071032, −13.7111231712, −12.8637449622, −12.5010608195, −11.6853720334, −11.1194727445, −10.2211164916, −9.61044720359, −8.96488622754, −8.39242173597, −7.21496298877, −6.53071100146, −5.96414046933, −4.15002132278, −3.04077030111, 3.04077030111, 4.15002132278, 5.96414046933, 6.53071100146, 7.21496298877, 8.39242173597, 8.96488622754, 9.61044720359, 10.2211164916, 11.1194727445, 11.6853720334, 12.5010608195, 12.8637449622, 13.7111231712, 14.4322071032, 15.3332748645, 15.7654481631, 16.0517876805, 16.7376346664, 17.3762435820, 18.0317193227, 18.5939242709, 19.0223801228, 19.5268623180

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