Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.225 + 0.974i$
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 + (−0.504 + 2.17i)5-s + 3.91i·7-s − 9-s + 3.46·11-s − 6.46i·13-s + (−2.17 − 0.504i)15-s + 0.603i·17-s − 0.547·19-s − 3.91·21-s − 7.27i·23-s + (−4.49 − 2.19i)25-s i·27-s − 5.89·29-s − 4.98·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.225 + 0.974i)5-s + 1.48i·7-s − 0.333·9-s + 1.04·11-s − 1.79i·13-s + (−0.562 − 0.130i)15-s + 0.146i·17-s − 0.125·19-s − 0.854·21-s − 1.51i·23-s + (−0.898 − 0.439i)25-s − 0.192i·27-s − 1.09·29-s − 0.895·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.225 + 0.974i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.225 + 0.974i)$
$L(1)$  $\approx$  $0.03206185606$
$L(\frac12)$  $\approx$  $0.03206185606$
$L(\frac{3}{2})$   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 \( 1 + (0.504 - 2.17i)T \)
67 \( 1 - iT \)
good7 \( 1 - 3.91iT - 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 6.46iT - 13T^{2} \)
17 \( 1 - 0.603iT - 17T^{2} \)
19 \( 1 + 0.547T + 19T^{2} \)
23 \( 1 + 7.27iT - 23T^{2} \)
29 \( 1 + 5.89T + 29T^{2} \)
31 \( 1 + 4.98T + 31T^{2} \)
37 \( 1 + 2.50iT - 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 - 3.38iT - 43T^{2} \)
47 \( 1 - 5.74iT - 47T^{2} \)
53 \( 1 + 2.06iT - 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 8.87T + 61T^{2} \)
71 \( 1 - 3.83T + 71T^{2} \)
73 \( 1 + 15.6iT - 73T^{2} \)
79 \( 1 + 8.63T + 79T^{2} \)
83 \( 1 - 3.96iT - 83T^{2} \)
89 \( 1 + 8.55T + 89T^{2} \)
97 \( 1 - 6.10iT - 97T^{2} \)
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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

−8.304893750317122621317875146525, −7.62252979646997935484874528239, −6.58151504729851726811614251934, −5.98012426149774034361216759470, −5.39710458995509635389171464443, −4.40273025330957255399073033877, −3.36309532670098148611032725691, −2.91952678833599250740109641560, −1.90740085409058175520204839999, −0.008830706586360142698772395959, 1.42114997248082188888658922500, 1.68018121357819702099534236578, 3.58916621463473059549554195582, 3.99385398686453141558500985981, 4.77613519209823935185244316607, 5.72220747279880841932244337594, 6.74446379190563583573464875712, 7.13548760510835966250725198645, 7.77146203756904968256133395412, 8.727479327100585972842822690377

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