Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.992 + 0.119i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.19 − 0.755i)2-s + (0.859 − 1.80i)4-s − 3.16i·5-s − 7-s + (−0.336 − 2.80i)8-s + (−2.39 − 3.78i)10-s − 5.44i·11-s + 3.61i·13-s + (−1.19 + 0.755i)14-s + (−2.52 − 3.10i)16-s − 3.27·17-s + 3.20i·19-s + (−5.72 − 2.72i)20-s + (−4.11 − 6.51i)22-s − 0.673·23-s + ⋯
L(s)  = 1  + (0.845 − 0.534i)2-s + (0.429 − 0.903i)4-s − 1.41i·5-s − 0.377·7-s + (−0.119 − 0.992i)8-s + (−0.757 − 1.19i)10-s − 1.64i·11-s + 1.00i·13-s + (−0.319 + 0.201i)14-s + (−0.630 − 0.775i)16-s − 0.794·17-s + 0.735i·19-s + (−1.28 − 0.608i)20-s + (−0.877 − 1.38i)22-s − 0.140·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.992 + 0.119i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ -0.992 + 0.119i)$
$L(1)$  $\approx$  $2.193565406$
$L(\frac12)$  $\approx$  $2.193565406$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.755i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 3.16iT - 5T^{2} \)
11 \( 1 + 5.44iT - 11T^{2} \)
13 \( 1 - 3.61iT - 13T^{2} \)
17 \( 1 + 3.27T + 17T^{2} \)
19 \( 1 - 3.20iT - 19T^{2} \)
23 \( 1 + 0.673T + 23T^{2} \)
29 \( 1 - 2.85iT - 29T^{2} \)
31 \( 1 - 3.71T + 31T^{2} \)
37 \( 1 + 11.9iT - 37T^{2} \)
41 \( 1 - 7.44T + 41T^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 - 4.06T + 47T^{2} \)
53 \( 1 - 0.291iT - 53T^{2} \)
59 \( 1 - 0.0209iT - 59T^{2} \)
61 \( 1 + 5.34iT - 61T^{2} \)
67 \( 1 + 6.20iT - 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 - 1.35T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 13.6iT - 83T^{2} \)
89 \( 1 - 5.93T + 89T^{2} \)
97 \( 1 - 9.60T + 97T^{2} \)
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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

−9.049169778547849798081972117318, −8.627002534402446439384452238419, −7.41270262843343530154694316012, −6.13739376553276861538600949200, −5.84121314544139093765335124323, −4.69027784343016115771568144192, −4.10778940386051735014223669123, −3.07650483960987598229361905007, −1.76524561892432315484268807150, −0.62493901999726350076248227045, 2.34054952403822952307172964941, 2.88720069476816821133681898224, 4.00099447003491729689696995403, 4.82884419265733867747847259311, 5.90740223398885006846269145271, 6.72857631815366578522049016129, 7.15075878081515113124404007860, 7.87189931175719888572244989631, 8.973773390822871323190997608839, 10.12326594806568025646488299183

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