Properties

Degree $4$
Conductor $1109$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 5-s − 6-s + 7-s + 8-s − 3·9-s + 10-s + 5·11-s − 12-s − 7·13-s − 14-s − 15-s − 16-s − 2·17-s + 3·18-s + 2·19-s + 20-s + 21-s − 5·22-s + 6·23-s + 24-s − 2·25-s + 7·26-s − 4·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 1.94·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.458·19-s + 0.223·20-s + 0.218·21-s − 1.06·22-s + 1.25·23-s + 0.204·24-s − 2/5·25-s + 1.37·26-s − 0.769·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1109\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1109} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1109,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3820859774\)
\(L(\frac12)\) \(\approx\) \(0.3820859774\)
\(L(\frac{3}{2})\) 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)$
bad1109$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 60 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
5$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$D_{4}$ \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T - 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + T + 47 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 3 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 7 T + 52 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$D_{4}$ \( 1 + 5 T + 90 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 22 T + 264 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
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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

−19.5668973954, −19.3951229570, −18.9902525979, −17.9964816252, −17.5062868341, −17.2577837662, −16.7831657294, −15.9158558267, −15.0456568694, −14.7149635447, −14.1550079654, −13.7697677175, −12.7589448000, −12.0130052529, −11.5874409074, −10.9070009166, −9.66881435554, −9.43797063076, −8.76277326868, −8.19769246356, −7.37895581281, −6.52506111333, −5.14556386773, −4.27546771886, −2.82389950240, 2.82389950240, 4.27546771886, 5.14556386773, 6.52506111333, 7.37895581281, 8.19769246356, 8.76277326868, 9.43797063076, 9.66881435554, 10.9070009166, 11.5874409074, 12.0130052529, 12.7589448000, 13.7697677175, 14.1550079654, 14.7149635447, 15.0456568694, 15.9158558267, 16.7831657294, 17.2577837662, 17.5062868341, 17.9964816252, 18.9902525979, 19.3951229570, 19.5668973954

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