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  − 5.20e5·2-s + 3.87e8·3-s + 1.33e11·4-s − 2.63e12·5-s − 2.01e14·6-s + 8.34e15·7-s + 1.97e15·8-s + 1.50e17·9-s + 1.37e18·10-s − 2.00e19·11-s + 5.17e19·12-s − 1.05e20·13-s − 4.34e21·14-s − 1.02e21·15-s − 1.93e22·16-s + 1.77e22·17-s − 7.81e22·18-s + 4.00e23·19-s − 3.52e23·20-s + 3.23e24·21-s + 1.04e25·22-s − 1.07e25·23-s + 7.64e23·24-s − 6.58e25·25-s + 5.48e25·26-s + 5.81e25·27-s + 1.11e27·28-s + ⋯
L(s)  = 1  − 1.40·2-s + 0.577·3-s + 0.972·4-s − 0.308·5-s − 0.810·6-s + 1.93·7-s + 0.0387·8-s + 0.333·9-s + 0.433·10-s − 1.08·11-s + 0.561·12-s − 0.259·13-s − 2.71·14-s − 0.178·15-s − 1.02·16-s + 0.305·17-s − 0.468·18-s + 0.882·19-s − 0.300·20-s + 1.11·21-s + 1.52·22-s − 0.692·23-s + 0.0223·24-s − 0.904·25-s + 0.364·26-s + 0.192·27-s + 1.88·28-s + ⋯

Functional equation

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

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  =  $(2,\ 3,\ (\ :37/2),\ 1)$
$L(19)$  $\approx$  $1.302333512$
$L(\frac12)$  $\approx$  $1.302333512$
$L(\frac{39}{2})$   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 \( 1 + 5.20e5T + 1.37e11T^{2} \)
5 \( 1 + 2.63e12T + 7.27e25T^{2} \)
7 \( 1 - 8.34e15T + 1.85e31T^{2} \)
11 \( 1 + 2.00e19T + 3.40e38T^{2} \)
13 \( 1 + 1.05e20T + 1.64e41T^{2} \)
17 \( 1 - 1.77e22T + 3.36e45T^{2} \)
19 \( 1 - 4.00e23T + 2.06e47T^{2} \)
23 \( 1 + 1.07e25T + 2.42e50T^{2} \)
29 \( 1 - 1.59e27T + 1.28e54T^{2} \)
31 \( 1 - 4.20e27T + 1.51e55T^{2} \)
37 \( 1 - 1.71e29T + 1.05e58T^{2} \)
41 \( 1 + 3.95e29T + 4.70e59T^{2} \)
43 \( 1 - 3.48e29T + 2.74e60T^{2} \)
47 \( 1 + 2.42e30T + 7.37e61T^{2} \)
53 \( 1 + 1.34e32T + 6.28e63T^{2} \)
59 \( 1 - 1.06e33T + 3.32e65T^{2} \)
61 \( 1 + 7.04e32T + 1.14e66T^{2} \)
67 \( 1 - 7.19e33T + 3.67e67T^{2} \)
71 \( 1 - 2.35e34T + 3.13e68T^{2} \)
73 \( 1 - 2.69e33T + 8.76e68T^{2} \)
79 \( 1 - 2.20e34T + 1.63e70T^{2} \)
83 \( 1 + 2.35e35T + 1.01e71T^{2} \)
89 \( 1 - 1.13e35T + 1.34e72T^{2} \)
97 \( 1 - 8.44e36T + 3.24e73T^{2} \)
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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

−17.57628386085364901798925449844, −15.70901808993541238107255103059, −14.04543302166871492725100922970, −11.48591441628306933787316764005, −10.03059963989914318847138277500, −8.201514534597191258071216342069, −7.76577579717934416815426708984, −4.74024898788082779304229807135, −2.24733023595259763119491346067, −0.936580817219163961350976004255, 0.936580817219163961350976004255, 2.24733023595259763119491346067, 4.74024898788082779304229807135, 7.76577579717934416815426708984, 8.201514534597191258071216342069, 10.03059963989914318847138277500, 11.48591441628306933787316764005, 14.04543302166871492725100922970, 15.70901808993541238107255103059, 17.57628386085364901798925449844

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