Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.888 - 0.458i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−2 − 1.73i)7-s + (−2.5 + 4.33i)11-s + (2.5 − 4.33i)13-s + (1.5 + 2.59i)17-s + (0.5 − 0.866i)19-s + (−1.5 − 2.59i)23-s + (2 − 3.46i)25-s + (−0.5 − 0.866i)29-s + (−0.499 + 2.59i)35-s + (−1.5 + 2.59i)37-s + (−2.5 + 4.33i)41-s + (−0.5 − 0.866i)43-s + (1.00 + 6.92i)49-s + (−4.5 − 7.79i)53-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.755 − 0.654i)7-s + (−0.753 + 1.30i)11-s + (0.693 − 1.20i)13-s + (0.363 + 0.630i)17-s + (0.114 − 0.198i)19-s + (−0.312 − 0.541i)23-s + (0.400 − 0.692i)25-s + (−0.0928 − 0.160i)29-s + (−0.0845 + 0.439i)35-s + (−0.246 + 0.427i)37-s + (−0.390 + 0.676i)41-s + (−0.0762 − 0.132i)43-s + (0.142 + 0.989i)49-s + (−0.618 − 1.07i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.888 - 0.458i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (2305, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.888 - 0.458i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.089451762921215009046955054423, −7.72658652821891308715503167176, −6.76692288238575305659273914060, −6.08773006047263752108762735136, −5.10467047115432164762679956830, −4.39143190083494020368950107795, −3.49376852939387318069389909272, −2.62467224757714468248644558341, −1.26860482922731089060873685054, 0, 1.61731925739972133106433759324, 2.97913221524824568441031630067, 3.31691634694299195356058096122, 4.43599268181982527574138730490, 5.64881878332391680566977770496, 5.94465366090596948038857765762, 6.93528039698930848788789197911, 7.56980103187178263220904642885, 8.600423653229840408808767033313

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