Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.943 + 0.332i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.29 − 0.564i)2-s + (1.36 − 1.46i)4-s + 1.53i·5-s + 7-s + (0.939 − 2.66i)8-s + (0.866 + 1.98i)10-s − 2.28i·11-s + 7.10i·13-s + (1.29 − 0.564i)14-s + (−0.288 − 3.98i)16-s + 6.81·17-s − 1.60i·19-s + (2.24 + 2.08i)20-s + (−1.28 − 2.96i)22-s + 1.16·23-s + ⋯
L(s)  = 1  + (0.916 − 0.399i)2-s + (0.681 − 0.732i)4-s + 0.685i·5-s + 0.377·7-s + (0.332 − 0.943i)8-s + (0.273 + 0.628i)10-s − 0.688i·11-s + 1.97i·13-s + (0.346 − 0.150i)14-s + (−0.0720 − 0.997i)16-s + 1.65·17-s − 0.368i·19-s + (0.502 + 0.467i)20-s + (−0.274 − 0.631i)22-s + 0.243·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.943 + 0.332i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.943 + 0.332i)\)
\(L(1)\)  \(\approx\)  \(3.327286452\)
\(L(\frac12)\)  \(\approx\)  \(3.327286452\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.29 + 0.564i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 1.53iT - 5T^{2} \)
11 \( 1 + 2.28iT - 11T^{2} \)
13 \( 1 - 7.10iT - 13T^{2} \)
17 \( 1 - 6.81T + 17T^{2} \)
19 \( 1 + 1.60iT - 19T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 - 4.07iT - 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + 7.04iT - 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 0.344iT - 43T^{2} \)
47 \( 1 - 3.10T + 47T^{2} \)
53 \( 1 + 5.66iT - 53T^{2} \)
59 \( 1 + 1.38iT - 59T^{2} \)
61 \( 1 - 5.50iT - 61T^{2} \)
67 \( 1 - 8.35iT - 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 5.95T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 2.60T + 97T^{2} \)
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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

−9.515548968610615796314645226711, −8.824992620846904273324989206297, −7.42220561298572525390140872454, −6.99101988334087611794188004054, −6.00089233924071719692909266973, −5.28904726049098307664961509265, −4.21757326832454750520608967382, −3.46083729100084126356042369490, −2.45767283303233843386423991301, −1.32315100962801392103107845013, 1.22163121867050625188809166822, 2.69835142783511729620817424106, 3.61139490301487428644640236112, 4.65791267212930267683980580722, 5.43366777658361142925639776991, 5.85752662954043823863637564764, 7.21397726311366402704926043338, 7.84914498021480121830518494466, 8.332118830015968436557539109379, 9.528325434064419646018359409780

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