Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.999 - 0.0238i$
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 + (−2.23 − 0.0533i)5-s + 4.50i·7-s − 9-s + 0.971·11-s − 1.87i·13-s + (0.0533 − 2.23i)15-s + 5.03i·17-s + 1.77·19-s − 4.50·21-s + 5.44i·23-s + (4.99 + 0.238i)25-s i·27-s + 3.85·29-s + 5.05·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.999 − 0.0238i)5-s + 1.70i·7-s − 0.333·9-s + 0.292·11-s − 0.519i·13-s + (0.0137 − 0.577i)15-s + 1.21i·17-s + 0.407·19-s − 0.983·21-s + 1.13i·23-s + (0.998 + 0.0477i)25-s − 0.192i·27-s + 0.715·29-s + 0.907·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0238i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0238i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.999 - 0.0238i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.999 - 0.0238i)$
$L(1)$  $\approx$  $1.151960322$
$L(\frac12)$  $\approx$  $1.151960322$
$L(\frac{3}{2})$   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 \( 1 + (2.23 + 0.0533i)T \)
67 \( 1 - iT \)
good7 \( 1 - 4.50iT - 7T^{2} \)
11 \( 1 - 0.971T + 11T^{2} \)
13 \( 1 + 1.87iT - 13T^{2} \)
17 \( 1 - 5.03iT - 17T^{2} \)
19 \( 1 - 1.77T + 19T^{2} \)
23 \( 1 - 5.44iT - 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 - 5.05T + 31T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 - 5.79T + 41T^{2} \)
43 \( 1 - 8.65iT - 43T^{2} \)
47 \( 1 + 1.27iT - 47T^{2} \)
53 \( 1 + 4.85iT - 53T^{2} \)
59 \( 1 - 0.801T + 59T^{2} \)
61 \( 1 + 9.97T + 61T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 + 16.8iT - 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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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

−8.591566691077703156193129409823, −8.330127226682971899281570458145, −7.56184573534612371492340186870, −6.38865361833447126871778561834, −5.85126852144442796215940688747, −5.00854852376213523787368158026, −4.32030346100750131776520579132, −3.28213919538464031719268666191, −2.80035285274793040042100298575, −1.40468821586827398367136628100, 0.40310591457020646475558901802, 1.07329176963190896397977778097, 2.54840865881914459902347325652, 3.50003389635570127384856585933, 4.31272636961553581900640899676, 4.76082270275259615024077108380, 6.09158293623378268889074323434, 6.99145553273647535855593196476, 7.23467889221038453880464376567, 7.85005705103318610642126506187

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