Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.380 - 0.924i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.88·5-s + (2.62 − 0.339i)7-s − 1.48·11-s + 2·13-s + 3.76i·17-s + 0.679i·19-s − 5.20i·23-s + 10.0·25-s + 7.03i·29-s − 6.42·31-s + (−10.1 + 1.32i)35-s − 8.71i·37-s − 3.16i·41-s − 8.20·43-s + 9.88·47-s + ⋯
L(s)  = 1  − 1.73·5-s + (0.991 − 0.128i)7-s − 0.446·11-s + 0.554·13-s + 0.913i·17-s + 0.155i·19-s − 1.08i·23-s + 2.01·25-s + 1.30i·29-s − 1.15·31-s + (−1.72 + 0.223i)35-s − 1.43i·37-s − 0.493i·41-s − 1.25·43-s + 1.44·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.380 - 0.924i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.380 - 0.924i)$
$L(1)$  $\approx$  $0.7139643403$
$L(\frac12)$  $\approx$  $0.7139643403$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.339i)T \)
good5 \( 1 + 3.88T + 5T^{2} \)
11 \( 1 + 1.48T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 3.76iT - 17T^{2} \)
19 \( 1 - 0.679iT - 19T^{2} \)
23 \( 1 + 5.20iT - 23T^{2} \)
29 \( 1 - 7.03iT - 29T^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 + 8.71iT - 37T^{2} \)
41 \( 1 + 3.16iT - 41T^{2} \)
43 \( 1 + 8.20T + 43T^{2} \)
47 \( 1 - 9.88T + 47T^{2} \)
53 \( 1 + 6.42iT - 53T^{2} \)
59 \( 1 - 7.76iT - 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 8.81T + 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 - 4.44iT - 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 - 9.88iT - 97T^{2} \)
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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

−8.558297567428485891902273773447, −7.931653412889356054164953765237, −7.40627787191124486928794754007, −6.68583958516277879491271239588, −5.53722788089917803456227714201, −4.83153893068856587989572243410, −3.94811958692293021224285979296, −3.59521761066029935272673248328, −2.27953161796830063214484788464, −1.05194115337305696251236064439, 0.24594965859372605432813499733, 1.50502126150712678040164428497, 2.84278574124399048067981246442, 3.62883057054776796958481335413, 4.45286612658444236685934761393, 5.00065610878957687063820178336, 5.92807987965008552845067780536, 7.07446725541115516040152731443, 7.60776818395033584073224435893, 8.081993630367176917497905264182

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