Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.128 - 0.991i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 0.0608i)2-s + (1.99 + 0.172i)4-s + 3.11i·5-s + 7-s + (2.80 + 0.364i)8-s + (−0.189 + 4.39i)10-s + 2.37i·11-s + 1.09i·13-s + (1.41 + 0.0608i)14-s + (3.94 + 0.685i)16-s − 3.69·17-s + 1.08i·19-s + (−0.535 + 6.20i)20-s + (−0.144 + 3.35i)22-s − 4.87·23-s + ⋯
L(s)  = 1  + (0.999 + 0.0430i)2-s + (0.996 + 0.0860i)4-s + 1.39i·5-s + 0.377·7-s + (0.991 + 0.128i)8-s + (−0.0599 + 1.39i)10-s + 0.715i·11-s + 0.302i·13-s + (0.377 + 0.0162i)14-s + (0.985 + 0.171i)16-s − 0.895·17-s + 0.248i·19-s + (−0.119 + 1.38i)20-s + (−0.0308 + 0.715i)22-s − 1.01·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.128 - 0.991i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.128 - 0.991i)\)
\(L(1)\)  \(\approx\)  \(3.310787921\)
\(L(\frac12)\)  \(\approx\)  \(3.310787921\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.41 - 0.0608i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 3.11iT - 5T^{2} \)
11 \( 1 - 2.37iT - 11T^{2} \)
13 \( 1 - 1.09iT - 13T^{2} \)
17 \( 1 + 3.69T + 17T^{2} \)
19 \( 1 - 1.08iT - 19T^{2} \)
23 \( 1 + 4.87T + 23T^{2} \)
29 \( 1 - 1.59iT - 29T^{2} \)
31 \( 1 - 7.45T + 31T^{2} \)
37 \( 1 + 4.61iT - 37T^{2} \)
41 \( 1 - 0.0380T + 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + 0.337T + 47T^{2} \)
53 \( 1 + 8.14iT - 53T^{2} \)
59 \( 1 - 15.0iT - 59T^{2} \)
61 \( 1 - 5.53iT - 61T^{2} \)
67 \( 1 - 7.70iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 - 5.18T + 79T^{2} \)
83 \( 1 + 0.138iT - 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 5.30T + 97T^{2} \)
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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

−10.13656280071634693074472808970, −8.746947861163982099607443948146, −7.68372704987874567898275485587, −7.03648019027653670954255643356, −6.45816682363444488853482831610, −5.58595303047859977246065525098, −4.48059716776812795139517267346, −3.78682007193919109992567967118, −2.64632515645357073733632713895, −1.95397798330881496848607122926, 0.944294235806157862491519061621, 2.14842580518086501902279321958, 3.37755243220455679981560069511, 4.55431241867445267619727470980, 4.82993708827937049677814920291, 5.91138503701999075230325669139, 6.51144431459236386368956958887, 7.891742415563383905528593398919, 8.256790750930348662206247940752, 9.248502378972483755107162078404

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