Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.600 - 0.799i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (1.34 − 1.78i)5-s − 0.521i·7-s − 9-s − 3.11·11-s + 5.31i·13-s + (1.78 + 1.34i)15-s − 5.37i·17-s + 0.478·19-s + 0.521·21-s + 9.04i·23-s + (−1.39 − 4.80i)25-s i·27-s + 3.80·29-s + 3.77·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.600 − 0.799i)5-s − 0.197i·7-s − 0.333·9-s − 0.938·11-s + 1.47i·13-s + (0.461 + 0.346i)15-s − 1.30i·17-s + 0.109·19-s + 0.113·21-s + 1.88i·23-s + (−0.278 − 0.960i)25-s − 0.192i·27-s + 0.707·29-s + 0.678·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.600 - 0.799i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.600 - 0.799i)$
$L(1)$  $\approx$  $1.811247796$
$L(\frac12)$  $\approx$  $1.811247796$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 + (-1.34 + 1.78i)T \)
67 \( 1 - iT \)
good7 \( 1 + 0.521iT - 7T^{2} \)
11 \( 1 + 3.11T + 11T^{2} \)
13 \( 1 - 5.31iT - 13T^{2} \)
17 \( 1 + 5.37iT - 17T^{2} \)
19 \( 1 - 0.478T + 19T^{2} \)
23 \( 1 - 9.04iT - 23T^{2} \)
29 \( 1 - 3.80T + 29T^{2} \)
31 \( 1 - 3.77T + 31T^{2} \)
37 \( 1 + 5.87iT - 37T^{2} \)
41 \( 1 - 5.50T + 41T^{2} \)
43 \( 1 + 2.92iT - 43T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 - 13.8iT - 53T^{2} \)
59 \( 1 + 2.17T + 59T^{2} \)
61 \( 1 + 7.47T + 61T^{2} \)
71 \( 1 + 4.15T + 71T^{2} \)
73 \( 1 - 8.31iT - 73T^{2} \)
79 \( 1 - 6.27T + 79T^{2} \)
83 \( 1 - 5.47iT - 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 18.8iT - 97T^{2} \)
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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

−8.859532322660821329560434066139, −7.74607414027992645510574276525, −7.25829785286431857107454062538, −6.11878273893715520746025258858, −5.52176545339373535880448105571, −4.70660977348337236032235167568, −4.26156189652818864544118729755, −3.03973002788348626122981743511, −2.16261011455400297016162657699, −0.986125597329853064808572789622, 0.60274293960759654977893877744, 2.01437506425617340313570322868, 2.71680646348798278940140222482, 3.39407138339606113237251018612, 4.72868434964370738134756240688, 5.54104573785958133167429419016, 6.23032014787069342914990236211, 6.69869991197752920954493281366, 7.76013856225658661879121600738, 8.161714310622497239750134638518

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