Properties

Degree 2
Conductor $ 3 \cdot 37 $
Sign $0.621 + 0.783i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−1 − 1.73i)19-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.499 − 0.866i)28-s − 31-s + 0.999·36-s + 37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−1 − 1.73i)19-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.499 − 0.866i)28-s − 31-s + 0.999·36-s + 37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(111\)    =    \(3 \cdot 37\)
\( \varepsilon \)  =  $0.621 + 0.783i$
motivic weight  =  \(0\)
character  :  $\chi_{111} (47, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 111,\ (\ :0),\ 0.621 + 0.783i)$
$L(\frac{1}{2})$  $\approx$  $0.4872960951$
$L(\frac12)$  $\approx$  $0.4872960951$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;37\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;37\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−13.62304570253849326090067990079, −12.87351548979488832581956316560, −11.50359384581484360434866422988, −10.96748126909624518403723367049, −9.323550309234632306437585417334, −8.463172696586380663441779432537, −6.86328204660327443830310453966, −5.81220143111909919666559615863, −4.74283713726553977917887541470, −1.94869000785363089009706569365, 3.60018671529121378340731000668, 4.48691441166618421750396295135, 5.97703246902699662443433916656, 7.70723203883669501750919805383, 8.616652078435595282895870814145, 10.00763679155022500534844766937, 10.80389228668928613087785431902, 11.95902905251912013524634114598, 12.93352522765451202314856167685, 14.08859395282092172934470765459

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