Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.524 - 0.851i$
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 + (1.17 − 1.90i)5-s + 2.37i·7-s − 9-s − 2.77·11-s + 0.720i·13-s + (−1.90 − 1.17i)15-s − 3.36i·17-s − 5.08·19-s + 2.37·21-s + 6.99i·23-s + (−2.24 − 4.46i)25-s + i·27-s − 10.0·29-s + 7.55·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.524 − 0.851i)5-s + 0.898i·7-s − 0.333·9-s − 0.835·11-s + 0.199i·13-s + (−0.491 − 0.302i)15-s − 0.816i·17-s − 1.16·19-s + 0.518·21-s + 1.45i·23-s + (−0.449 − 0.893i)25-s + 0.192i·27-s − 1.87·29-s + 1.35·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.524 - 0.851i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.524 - 0.851i)$
$L(1)$  $\approx$  $1.188881413$
$L(\frac12)$  $\approx$  $1.188881413$
$L(\frac{3}{2})$   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 \( 1 + (-1.17 + 1.90i)T \)
67 \( 1 + iT \)
good7 \( 1 - 2.37iT - 7T^{2} \)
11 \( 1 + 2.77T + 11T^{2} \)
13 \( 1 - 0.720iT - 13T^{2} \)
17 \( 1 + 3.36iT - 17T^{2} \)
19 \( 1 + 5.08T + 19T^{2} \)
23 \( 1 - 6.99iT - 23T^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
31 \( 1 - 7.55T + 31T^{2} \)
37 \( 1 - 10.4iT - 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 7.57iT - 43T^{2} \)
47 \( 1 + 0.513iT - 47T^{2} \)
53 \( 1 + 2.32iT - 53T^{2} \)
59 \( 1 - 9.22T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 - 4.90iT - 73T^{2} \)
79 \( 1 + 2.46T + 79T^{2} \)
83 \( 1 - 7.91iT - 83T^{2} \)
89 \( 1 + 3.49T + 89T^{2} \)
97 \( 1 - 12.5iT - 97T^{2} \)
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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

−8.404965709057482150870003303456, −8.041378415766855612973351593063, −7.11642044326788913780034931704, −6.22739983301776460282251053009, −5.56759379693784364500503992452, −5.07095033011746742162980078348, −4.08483179159166243787318531133, −2.74831370362284546782774261482, −2.17266300010734905636648835108, −1.11870476454522354138933165296, 0.34818539364610526347647978357, 2.06782887316312477579950745751, 2.73634364546899680309050353904, 3.90580908092707435826060994226, 4.27015441468156921403761839581, 5.50599585672865093376582104657, 6.00959732485339861355219246247, 6.90087464604906641311709408751, 7.52202822788701494425115969795, 8.345726236056787045959444797936

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