Properties

Degree 2
Conductor 3
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 9·3-s + 4·4-s + 6·5-s − 54·6-s − 40·7-s + 168·8-s + 81·9-s − 36·10-s − 564·11-s + 36·12-s + 638·13-s + 240·14-s + 54·15-s − 1.13e3·16-s + 882·17-s − 486·18-s − 556·19-s + 24·20-s − 360·21-s + 3.38e3·22-s − 840·23-s + 1.51e3·24-s − 3.08e3·25-s − 3.82e3·26-s + 729·27-s − 160·28-s + ⋯
L(s)  = 1  − 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.107·5-s − 0.612·6-s − 0.308·7-s + 0.928·8-s + 1/3·9-s − 0.113·10-s − 1.40·11-s + 0.0721·12-s + 1.04·13-s + 0.327·14-s + 0.0619·15-s − 1.10·16-s + 0.740·17-s − 0.353·18-s − 0.353·19-s + 0.0134·20-s − 0.178·21-s + 1.49·22-s − 0.331·23-s + 0.535·24-s − 0.988·25-s − 1.11·26-s + 0.192·27-s − 0.0385·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :5/2),\ 1)$
$L(3)$  $\approx$  $0.560038$
$L(\frac12)$  $\approx$  $0.560038$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - p^{2} T \)
good2 \( 1 + 3 p T + p^{5} T^{2} \)
5 \( 1 - 6 T + p^{5} T^{2} \)
7 \( 1 + 40 T + p^{5} T^{2} \)
11 \( 1 + 564 T + p^{5} T^{2} \)
13 \( 1 - 638 T + p^{5} T^{2} \)
17 \( 1 - 882 T + p^{5} T^{2} \)
19 \( 1 + 556 T + p^{5} T^{2} \)
23 \( 1 + 840 T + p^{5} T^{2} \)
29 \( 1 - 4638 T + p^{5} T^{2} \)
31 \( 1 - 4400 T + p^{5} T^{2} \)
37 \( 1 + 2410 T + p^{5} T^{2} \)
41 \( 1 + 6870 T + p^{5} T^{2} \)
43 \( 1 - 9644 T + p^{5} T^{2} \)
47 \( 1 + 18672 T + p^{5} T^{2} \)
53 \( 1 - 33750 T + p^{5} T^{2} \)
59 \( 1 + 18084 T + p^{5} T^{2} \)
61 \( 1 - 39758 T + p^{5} T^{2} \)
67 \( 1 + 23068 T + p^{5} T^{2} \)
71 \( 1 + 4248 T + p^{5} T^{2} \)
73 \( 1 + 41110 T + p^{5} T^{2} \)
79 \( 1 - 21920 T + p^{5} T^{2} \)
83 \( 1 - 82452 T + p^{5} T^{2} \)
89 \( 1 + 94086 T + p^{5} T^{2} \)
97 \( 1 - 49442 T + p^{5} T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−26.31632323429552434476773195958, −25.49039651686391952170668490094, −23.27285159605986829052851077708, −21.00541571501829177153179926125, −19.26122659608083353163124384925, −18.01473709281664649971029857911, −15.98902792677346249735516659778, −13.48325735922020845694503419854, −10.16731596026753900427842670488, −8.203430050132529358009219641993, 8.203430050132529358009219641993, 10.16731596026753900427842670488, 13.48325735922020845694503419854, 15.98902792677346249735516659778, 18.01473709281664649971029857911, 19.26122659608083353163124384925, 21.00541571501829177153179926125, 23.27285159605986829052851077708, 25.49039651686391952170668490094, 26.31632323429552434476773195958

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