Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.16e5·2-s − 3.87e8·3-s + 1.29e11·4-s + 3.13e12·5-s − 2.00e14·6-s − 3.10e15·7-s − 3.99e15·8-s + 1.50e17·9-s + 1.62e18·10-s + 7.50e18·11-s − 5.02e19·12-s − 6.39e20·13-s − 1.60e21·14-s − 1.21e21·15-s − 1.98e22·16-s − 5.44e22·17-s + 7.75e22·18-s + 2.22e23·19-s + 4.06e23·20-s + 1.20e24·21-s + 3.87e24·22-s − 1.64e25·23-s + 1.54e24·24-s − 6.29e25·25-s − 3.30e26·26-s − 5.81e25·27-s − 4.03e26·28-s + ⋯
L(s)  = 1  + 1.39·2-s − 0.577·3-s + 0.943·4-s + 0.367·5-s − 0.804·6-s − 0.721·7-s − 0.0784·8-s + 0.333·9-s + 0.512·10-s + 0.406·11-s − 0.544·12-s − 1.57·13-s − 1.00·14-s − 0.212·15-s − 1.05·16-s − 0.939·17-s + 0.464·18-s + 0.491·19-s + 0.347·20-s + 0.416·21-s + 0.567·22-s − 1.05·23-s + 0.0452·24-s − 0.864·25-s − 2.19·26-s − 0.192·27-s − 0.681·28-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  =  1
Selberg data  =  $(2,\ 3,\ (\ :37/2),\ -1)$
$L(19)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + 3.87e8T \)
good2 \( 1 - 5.16e5T + 1.37e11T^{2} \)
5 \( 1 - 3.13e12T + 7.27e25T^{2} \)
7 \( 1 + 3.10e15T + 1.85e31T^{2} \)
11 \( 1 - 7.50e18T + 3.40e38T^{2} \)
13 \( 1 + 6.39e20T + 1.64e41T^{2} \)
17 \( 1 + 5.44e22T + 3.36e45T^{2} \)
19 \( 1 - 2.22e23T + 2.06e47T^{2} \)
23 \( 1 + 1.64e25T + 2.42e50T^{2} \)
29 \( 1 - 1.49e27T + 1.28e54T^{2} \)
31 \( 1 + 4.99e27T + 1.51e55T^{2} \)
37 \( 1 - 4.47e28T + 1.05e58T^{2} \)
41 \( 1 - 1.03e30T + 4.70e59T^{2} \)
43 \( 1 + 2.18e30T + 2.74e60T^{2} \)
47 \( 1 - 1.06e31T + 7.37e61T^{2} \)
53 \( 1 + 2.17e31T + 6.28e63T^{2} \)
59 \( 1 - 9.25e32T + 3.32e65T^{2} \)
61 \( 1 - 1.78e33T + 1.14e66T^{2} \)
67 \( 1 - 5.16e33T + 3.67e67T^{2} \)
71 \( 1 - 3.92e33T + 3.13e68T^{2} \)
73 \( 1 + 3.83e34T + 8.76e68T^{2} \)
79 \( 1 + 5.14e34T + 1.63e70T^{2} \)
83 \( 1 + 2.76e35T + 1.01e71T^{2} \)
89 \( 1 + 1.38e36T + 1.34e72T^{2} \)
97 \( 1 - 1.68e36T + 3.24e73T^{2} \)
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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

−15.95172347732059755024533561831, −14.35577260233084465821339173551, −12.93435173976311639946978298802, −11.79027033324656800733934773727, −9.710250737862231738263614748119, −6.78834759279786733870028128167, −5.52628826542779276966716319580, −4.14414859039483214881930830333, −2.42467700499391446064514779792, 0, 2.42467700499391446064514779792, 4.14414859039483214881930830333, 5.52628826542779276966716319580, 6.78834759279786733870028128167, 9.710250737862231738263614748119, 11.79027033324656800733934773727, 12.93435173976311639946978298802, 14.35577260233084465821339173551, 15.95172347732059755024533561831

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