Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.302 + 0.953i$
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 + (−0.676 + 2.13i)5-s + 4.23i·7-s − 9-s − 1.39·11-s + 6.36i·13-s + (−2.13 − 0.676i)15-s − 4.10i·17-s − 5.15·19-s − 4.23·21-s − 1.91i·23-s + (−4.08 − 2.88i)25-s i·27-s − 5.40·29-s + 0.479·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.302 + 0.953i)5-s + 1.59i·7-s − 0.333·9-s − 0.419·11-s + 1.76i·13-s + (−0.550 − 0.174i)15-s − 0.996i·17-s − 1.18·19-s − 0.923·21-s − 0.398i·23-s + (−0.817 − 0.576i)25-s − 0.192i·27-s − 1.00·29-s + 0.0861·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.302 + 0.953i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.302 + 0.953i)$
$L(1)$  $\approx$  $0.6850146004$
$L(\frac12)$  $\approx$  $0.6850146004$
$L(\frac{3}{2})$   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 \( 1 + (0.676 - 2.13i)T \)
67 \( 1 - iT \)
good7 \( 1 - 4.23iT - 7T^{2} \)
11 \( 1 + 1.39T + 11T^{2} \)
13 \( 1 - 6.36iT - 13T^{2} \)
17 \( 1 + 4.10iT - 17T^{2} \)
19 \( 1 + 5.15T + 19T^{2} \)
23 \( 1 + 1.91iT - 23T^{2} \)
29 \( 1 + 5.40T + 29T^{2} \)
31 \( 1 - 0.479T + 31T^{2} \)
37 \( 1 - 2.52iT - 37T^{2} \)
41 \( 1 - 4.14T + 41T^{2} \)
43 \( 1 - 0.604iT - 43T^{2} \)
47 \( 1 + 0.216iT - 47T^{2} \)
53 \( 1 - 2.89iT - 53T^{2} \)
59 \( 1 + 0.893T + 59T^{2} \)
61 \( 1 - 2.90T + 61T^{2} \)
71 \( 1 - 8.26T + 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + 0.636iT - 97T^{2} \)
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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

−9.079233494331647464647855685564, −8.370792957372999048094807266157, −7.48474075690998466287218617682, −6.59442548962907838088651766299, −6.12708749080061602199225754246, −5.18605310435184457507133699477, −4.43675953881673857922128197895, −3.57284375708984348270721504873, −2.50249777864993982852943016070, −2.16326571084433826558601202383, 0.22020202201143359717332049228, 0.987612001612939003008484699621, 2.06285640158895099132425695162, 3.45619563262720596305804715360, 3.99544185883370254133184932280, 4.94250427622602336897962697717, 5.68823695592223549149522735241, 6.49227024005021930920995650446, 7.49710289324735057304572925127, 7.87175219874757321654971218108

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