Properties

Degree 2
Conductor 3
Sign $1$
Motivic weight 37
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.21e5·2-s + 3.87e8·3-s − 3.37e10·4-s + 1.53e13·5-s + 1.24e14·6-s + 2.48e15·7-s − 5.51e16·8-s + 1.50e17·9-s + 4.93e18·10-s − 1.32e18·11-s − 1.30e19·12-s + 1.53e20·13-s + 8.00e20·14-s + 5.94e21·15-s − 1.31e22·16-s + 1.04e23·17-s + 4.83e22·18-s − 6.96e23·19-s − 5.17e23·20-s + 9.63e23·21-s − 4.25e23·22-s + 1.11e25·23-s − 2.13e25·24-s + 1.62e26·25-s + 4.93e25·26-s + 5.81e25·27-s − 8.39e25·28-s + ⋯
L(s)  = 1  + 0.868·2-s + 0.577·3-s − 0.245·4-s + 1.79·5-s + 0.501·6-s + 0.577·7-s − 1.08·8-s + 0.333·9-s + 1.56·10-s − 0.0716·11-s − 0.141·12-s + 0.378·13-s + 0.501·14-s + 1.03·15-s − 0.694·16-s + 1.80·17-s + 0.289·18-s − 1.53·19-s − 0.441·20-s + 0.333·21-s − 0.0622·22-s + 0.719·23-s − 0.624·24-s + 2.23·25-s + 0.328·26-s + 0.192·27-s − 0.141·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(37\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :37/2),\ 1)$
$L(19)$  $\approx$  $4.681073251$
$L(\frac12)$  $\approx$  $4.681073251$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 3.87e8T \)
good2 \( 1 - 3.21e5T + 1.37e11T^{2} \)
5 \( 1 - 1.53e13T + 7.27e25T^{2} \)
7 \( 1 - 2.48e15T + 1.85e31T^{2} \)
11 \( 1 + 1.32e18T + 3.40e38T^{2} \)
13 \( 1 - 1.53e20T + 1.64e41T^{2} \)
17 \( 1 - 1.04e23T + 3.36e45T^{2} \)
19 \( 1 + 6.96e23T + 2.06e47T^{2} \)
23 \( 1 - 1.11e25T + 2.42e50T^{2} \)
29 \( 1 - 1.43e27T + 1.28e54T^{2} \)
31 \( 1 - 3.90e26T + 1.51e55T^{2} \)
37 \( 1 + 8.53e28T + 1.05e58T^{2} \)
41 \( 1 - 4.51e29T + 4.70e59T^{2} \)
43 \( 1 + 2.04e30T + 2.74e60T^{2} \)
47 \( 1 + 3.80e30T + 7.37e61T^{2} \)
53 \( 1 - 2.08e31T + 6.28e63T^{2} \)
59 \( 1 + 1.01e33T + 3.32e65T^{2} \)
61 \( 1 + 1.04e33T + 1.14e66T^{2} \)
67 \( 1 - 2.33e33T + 3.67e67T^{2} \)
71 \( 1 - 1.08e33T + 3.13e68T^{2} \)
73 \( 1 + 3.23e34T + 8.76e68T^{2} \)
79 \( 1 + 2.35e35T + 1.63e70T^{2} \)
83 \( 1 + 1.58e35T + 1.01e71T^{2} \)
89 \( 1 + 3.16e35T + 1.34e72T^{2} \)
97 \( 1 - 6.89e36T + 3.24e73T^{2} \)
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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

−17.24173984718214311990961622725, −14.68437133106928544771240844599, −13.87638466013428231809020869359, −12.71939125736113886943321725806, −10.11908614912665704922563944539, −8.692730932069098079164912724810, −6.13792801550831622399659488265, −4.87642316070806856068622124168, −2.97816892620446375421952644036, −1.48277362283597403311602540510, 1.48277362283597403311602540510, 2.97816892620446375421952644036, 4.87642316070806856068622124168, 6.13792801550831622399659488265, 8.692730932069098079164912724810, 10.11908614912665704922563944539, 12.71939125736113886943321725806, 13.87638466013428231809020869359, 14.68437133106928544771240844599, 17.24173984718214311990961622725

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