Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.623 - 0.781i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 0.314i)2-s + (1.80 + 0.867i)4-s + 0.114i·5-s − 7-s + (2.21 + 1.76i)8-s + (−0.0360 + 0.157i)10-s + 0.412i·11-s − 1.73i·13-s + (−1.37 − 0.314i)14-s + (2.49 + 3.12i)16-s + 2.50·17-s + 6.85i·19-s + (−0.0994 + 0.206i)20-s + (−0.129 + 0.569i)22-s + 4.42·23-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + 0.0512i·5-s − 0.377·7-s + (0.781 + 0.623i)8-s + (−0.0114 + 0.0499i)10-s + 0.124i·11-s − 0.481i·13-s + (−0.368 − 0.0841i)14-s + (0.623 + 0.781i)16-s + 0.608·17-s + 1.57i·19-s + (−0.0222 + 0.0461i)20-s + (−0.0277 + 0.121i)22-s + 0.922·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.623 - 0.781i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.623 - 0.781i)$
$L(1)$  $\approx$  $3.226451260$
$L(\frac12)$  $\approx$  $3.226451260$
$L(\frac{3}{2})$   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 \( 1 + (-1.37 - 0.314i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 0.114iT - 5T^{2} \)
11 \( 1 - 0.412iT - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 - 2.50T + 17T^{2} \)
19 \( 1 - 6.85iT - 19T^{2} \)
23 \( 1 - 4.42T + 23T^{2} \)
29 \( 1 - 1.85iT - 29T^{2} \)
31 \( 1 - 5.60T + 31T^{2} \)
37 \( 1 - 4.39iT - 37T^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
43 \( 1 - 4.35iT - 43T^{2} \)
47 \( 1 + 7.23T + 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + 4.25iT - 59T^{2} \)
61 \( 1 + 7.35iT - 61T^{2} \)
67 \( 1 - 6.25iT - 67T^{2} \)
71 \( 1 + 0.608T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 - 8.19T + 79T^{2} \)
83 \( 1 + 4.88iT - 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 3.42T + 97T^{2} \)
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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

−9.832532918830813483098129583102, −8.500122778035718132093925181802, −7.909605420048920116766436333991, −6.94764897453544247364312022563, −6.29066582945136719941249821872, −5.40823973366479813590720202520, −4.66481871645109617906995231575, −3.50088543859160581940171585071, −2.92289116719143621338743707449, −1.47036799834461974390045189593, 1.03309800476160501529880668075, 2.52489074222898405899176255712, 3.24040460787541937416965412906, 4.38806697658742136353772390896, 5.04907435233432898129991599785, 6.02090178170225344008219971531, 6.83391376413875589540960335668, 7.40356973093353945279132396941, 8.671768192420769410411357132489, 9.424949072753182318548092716294

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