Properties

Degree $2$
Conductor $39$
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 + 2·5-s − 6-s − 4·7-s − 3·8-s + 9-s + 2·10-s + 4·11-s + 12-s + 13-s − 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s − 2·20-s + 4·21-s + 4·22-s + 3·24-s − 25-s + 26-s − 27-s + 4·28-s − 10·29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s + 0.872·21-s + 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.85·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{39} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8266878506\)
\(L(\frac12)\) \(\approx\) \(0.8266878506\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.46044217768725, −18.42827149360469, −17.28491480085541, −16.41358335985935, −14.93804365278040, −13.67450328613746, −12.95684499975162, −11.85387650647455, −9.952802346640542, −9.192740554674130, −6.545565450427837, −5.654867802797051, −3.712054470240156, 3.712054470240156, 5.654867802797051, 6.545565450427837, 9.192740554674130, 9.952802346640542, 11.85387650647455, 12.95684499975162, 13.67450328613746, 14.93804365278040, 16.41358335985935, 17.28491480085541, 18.42827149360469, 19.46044217768725

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