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  − 1.48e5·2-s − 3.87e8·3-s − 1.15e11·4-s + 7.60e11·5-s + 5.74e13·6-s + 2.59e15·7-s + 3.75e16·8-s + 1.50e17·9-s − 1.12e17·10-s + 3.65e18·11-s + 4.47e19·12-s + 6.05e19·13-s − 3.84e20·14-s − 2.94e20·15-s + 1.03e22·16-s + 5.25e22·17-s − 2.22e22·18-s − 4.58e23·19-s − 8.78e22·20-s − 1.00e24·21-s − 5.42e23·22-s + 6.97e24·23-s − 1.45e25·24-s − 7.21e25·25-s − 8.97e24·26-s − 5.81e25·27-s − 2.99e26·28-s + ⋯
L(s)  = 1  − 0.400·2-s − 0.577·3-s − 0.839·4-s + 0.0891·5-s + 0.230·6-s + 0.601·7-s + 0.736·8-s + 0.333·9-s − 0.0356·10-s + 0.198·11-s + 0.484·12-s + 0.149·13-s − 0.240·14-s − 0.0514·15-s + 0.545·16-s + 0.906·17-s − 0.133·18-s − 1.01·19-s − 0.0749·20-s − 0.347·21-s − 0.0793·22-s + 0.448·23-s − 0.425·24-s − 0.992·25-s − 0.0597·26-s − 0.192·27-s − 0.505·28-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 \( 1 + 1.48e5T + 1.37e11T^{2} \)
5 \( 1 - 7.60e11T + 7.27e25T^{2} \)
7 \( 1 - 2.59e15T + 1.85e31T^{2} \)
11 \( 1 - 3.65e18T + 3.40e38T^{2} \)
13 \( 1 - 6.05e19T + 1.64e41T^{2} \)
17 \( 1 - 5.25e22T + 3.36e45T^{2} \)
19 \( 1 + 4.58e23T + 2.06e47T^{2} \)
23 \( 1 - 6.97e24T + 2.42e50T^{2} \)
29 \( 1 + 2.14e27T + 1.28e54T^{2} \)
31 \( 1 + 3.44e27T + 1.51e55T^{2} \)
37 \( 1 - 6.38e28T + 1.05e58T^{2} \)
41 \( 1 - 5.40e29T + 4.70e59T^{2} \)
43 \( 1 - 3.10e30T + 2.74e60T^{2} \)
47 \( 1 + 1.05e31T + 7.37e61T^{2} \)
53 \( 1 + 5.28e31T + 6.28e63T^{2} \)
59 \( 1 - 1.00e32T + 3.32e65T^{2} \)
61 \( 1 + 1.03e32T + 1.14e66T^{2} \)
67 \( 1 + 4.87e33T + 3.67e67T^{2} \)
71 \( 1 + 1.97e34T + 3.13e68T^{2} \)
73 \( 1 - 2.72e34T + 8.76e68T^{2} \)
79 \( 1 - 1.01e35T + 1.63e70T^{2} \)
83 \( 1 + 4.82e35T + 1.01e71T^{2} \)
89 \( 1 + 2.01e36T + 1.34e72T^{2} \)
97 \( 1 - 5.47e36T + 3.24e73T^{2} \)
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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

−16.73335103273078067407632864951, −14.56853926271641268879680697533, −12.87877178221131670774002527492, −11.01875720746523459183004418153, −9.419926469450289449399755792438, −7.77126072857203644714244660286, −5.60320726242482522463626928834, −4.11153485400818866616038938197, −1.48239938529818179884509771849, 0, 1.48239938529818179884509771849, 4.11153485400818866616038938197, 5.60320726242482522463626928834, 7.77126072857203644714244660286, 9.419926469450289449399755792438, 11.01875720746523459183004418153, 12.87877178221131670774002527492, 14.56853926271641268879680697533, 16.73335103273078067407632864951

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