Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.211 - 0.977i$
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 + (−0.471 − 2.18i)5-s + 1.05i·7-s − 9-s − 5.95·11-s − 5.75i·13-s + (2.18 − 0.471i)15-s + 4.60i·17-s + 4.92·19-s − 1.05·21-s − 5.41i·23-s + (−4.55 + 2.06i)25-s i·27-s − 2.23·29-s − 3.73·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.211 − 0.977i)5-s + 0.398i·7-s − 0.333·9-s − 1.79·11-s − 1.59i·13-s + (0.564 − 0.121i)15-s + 1.11i·17-s + 1.13·19-s − 0.230·21-s − 1.12i·23-s + (−0.910 + 0.412i)25-s − 0.192i·27-s − 0.415·29-s − 0.670·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.211 - 0.977i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.211 - 0.977i)$
$L(1)$  $\approx$  $0.7806714691$
$L(\frac12)$  $\approx$  $0.7806714691$
$L(\frac{3}{2})$   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 \( 1 + (0.471 + 2.18i)T \)
67 \( 1 - iT \)
good7 \( 1 - 1.05iT - 7T^{2} \)
11 \( 1 + 5.95T + 11T^{2} \)
13 \( 1 + 5.75iT - 13T^{2} \)
17 \( 1 - 4.60iT - 17T^{2} \)
19 \( 1 - 4.92T + 19T^{2} \)
23 \( 1 + 5.41iT - 23T^{2} \)
29 \( 1 + 2.23T + 29T^{2} \)
31 \( 1 + 3.73T + 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 + 6.81iT - 43T^{2} \)
47 \( 1 - 8.17iT - 47T^{2} \)
53 \( 1 - 1.55iT - 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 2.79T + 61T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 7.57iT - 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 - 1.38T + 89T^{2} \)
97 \( 1 - 2.06iT - 97T^{2} \)
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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

−8.612040079639179874552191866769, −7.83101034502525693929710111090, −7.71978566400882023231133283375, −6.10045579686454009048103720918, −5.46966303055369614873947182275, −5.08193403437783015287447677015, −4.19272870926742563917990064892, −3.16857122041584645541656729909, −2.46965134683655231174551980178, −0.964176813992616286973271055101, 0.25773321124527547176626827905, 1.83656075246468042866381571611, 2.65230567709251323940753188054, 3.41228097497390170948720538564, 4.42371075792677178927074530022, 5.40092288559423373939209645799, 6.03104745238556012082034926569, 7.14643697228983124629732990371, 7.42813976750217551092273821690, 7.77106300122895904148157744194

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