Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.896 - 0.443i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 − 0.885i)2-s + (0.430 + 1.95i)4-s − 2.99i·5-s + 7-s + (1.25 − 2.53i)8-s + (−2.64 + 3.29i)10-s − 2.13i·11-s + 0.665i·13-s + (−1.10 − 0.885i)14-s + (−3.62 + 1.68i)16-s − 7.04·17-s − 1.39i·19-s + (5.83 − 1.28i)20-s + (−1.89 + 2.35i)22-s − 0.184·23-s + ⋯
L(s)  = 1  + (−0.779 − 0.626i)2-s + (0.215 + 0.976i)4-s − 1.33i·5-s + 0.377·7-s + (0.443 − 0.896i)8-s + (−0.837 + 1.04i)10-s − 0.645i·11-s + 0.184i·13-s + (−0.294 − 0.236i)14-s + (−0.907 + 0.420i)16-s − 1.70·17-s − 0.320i·19-s + (1.30 − 0.287i)20-s + (−0.404 + 0.502i)22-s − 0.0384·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.896 - 0.443i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ -0.896 - 0.443i)$
$L(1)$  $\approx$  $0.5138152492$
$L(\frac12)$  $\approx$  $0.5138152492$
$L(\frac{3}{2})$   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 \( 1 + (1.10 + 0.885i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 2.99iT - 5T^{2} \)
11 \( 1 + 2.13iT - 11T^{2} \)
13 \( 1 - 0.665iT - 13T^{2} \)
17 \( 1 + 7.04T + 17T^{2} \)
19 \( 1 + 1.39iT - 19T^{2} \)
23 \( 1 + 0.184T + 23T^{2} \)
29 \( 1 + 1.27iT - 29T^{2} \)
31 \( 1 + 7.62T + 31T^{2} \)
37 \( 1 + 6.69iT - 37T^{2} \)
41 \( 1 - 0.274T + 41T^{2} \)
43 \( 1 - 2.63iT - 43T^{2} \)
47 \( 1 - 5.77T + 47T^{2} \)
53 \( 1 - 7.48iT - 53T^{2} \)
59 \( 1 + 9.03iT - 59T^{2} \)
61 \( 1 - 9.80iT - 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 + 7.93T + 71T^{2} \)
73 \( 1 + 2.80T + 73T^{2} \)
79 \( 1 - 7.98T + 79T^{2} \)
83 \( 1 - 6.76iT - 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 0.107T + 97T^{2} \)
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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

−9.015374317571599349138223425417, −8.538073841582788840997739489278, −7.71203723751092948621233848744, −6.79156701405614396570283501684, −5.63371882815744027386376615896, −4.58053399175942730091636688663, −3.92490800621889641085407979482, −2.50311697101688800312546076814, −1.47993262737150302691312382372, −0.25784233866177522986066337286, 1.78299137478929376053378037600, 2.68197884985359050680424174472, 4.10804494933926434593199897122, 5.17009206039228206764235238767, 6.16659749409490993394622681141, 6.93717881657636183140039208583, 7.31934210605546590226667598050, 8.308941668059120826337045518289, 9.063920676947973253957966257015, 9.933984129691751870657056597116

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