Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.472 + 0.881i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.497 − 1.32i)2-s + (−1.50 − 1.31i)4-s + 1.25i·5-s + 7-s + (−2.49 + 1.33i)8-s + (1.65 + 0.624i)10-s + 1.55i·11-s − 1.07i·13-s + (0.497 − 1.32i)14-s + (0.528 + 3.96i)16-s + 0.0158·17-s − 2.35i·19-s + (1.65 − 1.88i)20-s + (2.06 + 0.775i)22-s + 5.95·23-s + ⋯
L(s)  = 1  + (0.351 − 0.936i)2-s + (−0.752 − 0.658i)4-s + 0.560i·5-s + 0.377·7-s + (−0.881 + 0.472i)8-s + (0.524 + 0.197i)10-s + 0.469i·11-s − 0.297i·13-s + (0.133 − 0.353i)14-s + (0.132 + 0.991i)16-s + 0.00383·17-s − 0.539i·19-s + (0.369 − 0.421i)20-s + (0.439 + 0.165i)22-s + 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.472 + 0.881i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.472 + 0.881i)\)
\(L(1)\)  \(\approx\)  \(1.937169301\)
\(L(\frac12)\)  \(\approx\)  \(1.937169301\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.497 + 1.32i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 1.25iT - 5T^{2} \)
11 \( 1 - 1.55iT - 11T^{2} \)
13 \( 1 + 1.07iT - 13T^{2} \)
17 \( 1 - 0.0158T + 17T^{2} \)
19 \( 1 + 2.35iT - 19T^{2} \)
23 \( 1 - 5.95T + 23T^{2} \)
29 \( 1 - 0.469iT - 29T^{2} \)
31 \( 1 - 1.69T + 31T^{2} \)
37 \( 1 + 4.59iT - 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 - 1.97iT - 43T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 - 1.86iT - 53T^{2} \)
59 \( 1 - 8.54iT - 59T^{2} \)
61 \( 1 + 3.92iT - 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 + 4.22T + 71T^{2} \)
73 \( 1 - 6.51T + 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 - 8.88iT - 83T^{2} \)
89 \( 1 + 0.240T + 89T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
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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

−9.380270868469511840023450424102, −8.866125473792213025087459690925, −7.73466816838107609678760853923, −6.88918294538902403643225114284, −5.86900729770929159761925014397, −4.96995429022219438818236732770, −4.21478968576199596376086707794, −3.07740763956705648319826823714, −2.34931589217680610556810972879, −0.972475048940122119393343199632, 0.987187092972617374714131237451, 2.77651579076203750054839567300, 3.92584320305692319049559943886, 4.75160917051390578272831458805, 5.47315419668786012206945415593, 6.31429198564815188617484064941, 7.19835990629191644998485256200, 7.965181475313513311127001298871, 8.757278751518251929806823831233, 9.194068424852696993441778434516

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