Properties

Label 2-3-1.1-c7-0-0
Degree $2$
Conductor $3$
Sign $1$
Analytic cond. $0.937155$
Root an. cond. $0.968067$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 27·3-s − 92·4-s + 390·5-s − 162·6-s − 64·7-s − 1.32e3·8-s + 729·9-s + 2.34e3·10-s − 948·11-s + 2.48e3·12-s − 5.09e3·13-s − 384·14-s − 1.05e4·15-s + 3.85e3·16-s + 2.83e4·17-s + 4.37e3·18-s − 8.62e3·19-s − 3.58e4·20-s + 1.72e3·21-s − 5.68e3·22-s − 1.52e4·23-s + 3.56e4·24-s + 7.39e4·25-s − 3.05e4·26-s − 1.96e4·27-s + 5.88e3·28-s + ⋯
L(s)  = 1  + 0.530·2-s − 0.577·3-s − 0.718·4-s + 1.39·5-s − 0.306·6-s − 0.0705·7-s − 0.911·8-s + 1/3·9-s + 0.739·10-s − 0.214·11-s + 0.414·12-s − 0.643·13-s − 0.0374·14-s − 0.805·15-s + 0.235·16-s + 1.40·17-s + 0.176·18-s − 0.288·19-s − 1.00·20-s + 0.0407·21-s − 0.113·22-s − 0.262·23-s + 0.526·24-s + 0.946·25-s − 0.341·26-s − 0.192·27-s + 0.0506·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Analytic conductor: \(0.937155\)
Root analytic conductor: \(0.968067\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.066991478\)
\(L(\frac12)\) \(\approx\) \(1.066991478\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{3} T \)
good2 \( 1 - 3 p T + p^{7} T^{2} \)
5 \( 1 - 78 p T + p^{7} T^{2} \)
7 \( 1 + 64 T + p^{7} T^{2} \)
11 \( 1 + 948 T + p^{7} T^{2} \)
13 \( 1 + 5098 T + p^{7} T^{2} \)
17 \( 1 - 28386 T + p^{7} T^{2} \)
19 \( 1 + 8620 T + p^{7} T^{2} \)
23 \( 1 + 15288 T + p^{7} T^{2} \)
29 \( 1 - 36510 T + p^{7} T^{2} \)
31 \( 1 + 276808 T + p^{7} T^{2} \)
37 \( 1 - 268526 T + p^{7} T^{2} \)
41 \( 1 + 629718 T + p^{7} T^{2} \)
43 \( 1 - 685772 T + p^{7} T^{2} \)
47 \( 1 - 583296 T + p^{7} T^{2} \)
53 \( 1 + 428058 T + p^{7} T^{2} \)
59 \( 1 - 1306380 T + p^{7} T^{2} \)
61 \( 1 - 300662 T + p^{7} T^{2} \)
67 \( 1 + 507244 T + p^{7} T^{2} \)
71 \( 1 - 5560632 T + p^{7} T^{2} \)
73 \( 1 - 1369082 T + p^{7} T^{2} \)
79 \( 1 + 6913720 T + p^{7} T^{2} \)
83 \( 1 + 4376748 T + p^{7} T^{2} \)
89 \( 1 + 8528310 T + p^{7} T^{2} \)
97 \( 1 + 8826814 T + p^{7} T^{2} \)
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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

−25.54817686804717054683245675808, −23.77483993812190169977491498761, −22.25560509011893321716162430765, −21.25980235857850236814323488170, −18.38152053727722786631753109498, −17.06943179326853310395592745601, −14.30293639828087452883388129670, −12.73084105514744035285788350135, −9.803091174428125705004304039748, −5.55437325971837090122957089810, 5.55437325971837090122957089810, 9.803091174428125705004304039748, 12.73084105514744035285788350135, 14.30293639828087452883388129670, 17.06943179326853310395592745601, 18.38152053727722786631753109498, 21.25980235857850236814323488170, 22.25560509011893321716162430765, 23.77483993812190169977491498761, 25.54817686804717054683245675808

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