Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.879 - 0.476i$
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 + (1.96 − 1.06i)5-s − 4.32i·7-s − 9-s + 3.68·11-s + 4.58i·13-s + (1.06 + 1.96i)15-s + 7.89i·17-s + 4.22·19-s + 4.32·21-s + 1.04i·23-s + (2.73 − 4.18i)25-s i·27-s − 5.93·29-s + 5.66·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.879 − 0.476i)5-s − 1.63i·7-s − 0.333·9-s + 1.11·11-s + 1.27i·13-s + (0.274 + 0.507i)15-s + 1.91i·17-s + 0.969·19-s + 0.943·21-s + 0.217i·23-s + (0.546 − 0.837i)25-s − 0.192i·27-s − 1.10·29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.879 - 0.476i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.879 - 0.476i)$
$L(1)$  $\approx$  $2.501426300$
$L(\frac12)$  $\approx$  $2.501426300$
$L(\frac{3}{2})$   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 \( 1 + (-1.96 + 1.06i)T \)
67 \( 1 - iT \)
good7 \( 1 + 4.32iT - 7T^{2} \)
11 \( 1 - 3.68T + 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 - 7.89iT - 17T^{2} \)
19 \( 1 - 4.22T + 19T^{2} \)
23 \( 1 - 1.04iT - 23T^{2} \)
29 \( 1 + 5.93T + 29T^{2} \)
31 \( 1 - 5.66T + 31T^{2} \)
37 \( 1 - 11.2iT - 37T^{2} \)
41 \( 1 - 1.36T + 41T^{2} \)
43 \( 1 - 6.90iT - 43T^{2} \)
47 \( 1 - 0.227iT - 47T^{2} \)
53 \( 1 + 4.37iT - 53T^{2} \)
59 \( 1 - 6.83T + 59T^{2} \)
61 \( 1 - 4.41T + 61T^{2} \)
71 \( 1 + 6.97T + 71T^{2} \)
73 \( 1 + 1.49iT - 73T^{2} \)
79 \( 1 + 5.16T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 - 2.71T + 89T^{2} \)
97 \( 1 + 8.15iT - 97T^{2} \)
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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

−8.597044208224383534118366563821, −7.87219954263977055119454388185, −6.77168343041298546325325420799, −6.44864557175891020496031822188, −5.52007971364372864820088890132, −4.41725981581761549995302656847, −4.15974604820896329885924733797, −3.28947109565025211708774302307, −1.71749510121107121110451758835, −1.16053937245296066648801403159, 0.818494526792188873950793091486, 2.09481271452412841475475105986, 2.68832337236500103396630635852, 3.41714729425995173976535338507, 4.97445864874888956462198503641, 5.67000471941090705057851228402, 5.94326831293843505378002509220, 6.99874358542033789996393132756, 7.45547982046485021128662941936, 8.519158483123005920008402010386

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