Properties

Degree $2$
Conductor $3024$
Sign $0.907 - 0.420i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.43·5-s + (1.83 + 1.90i)7-s + 4.40·11-s + (1.49 − 2.58i)13-s + (−0.542 + 0.939i)17-s + (3.74 + 6.48i)19-s − 4.32·23-s + 6.80·25-s + (−1.68 − 2.91i)29-s + (4.68 + 8.11i)31-s + (6.31 + 6.53i)35-s + (−2.50 − 4.34i)37-s + (1.20 − 2.08i)41-s + (−3.31 − 5.74i)43-s + (−1.50 + 2.60i)47-s + ⋯
L(s)  = 1  + 1.53·5-s + (0.695 + 0.718i)7-s + 1.32·11-s + (0.414 − 0.717i)13-s + (−0.131 + 0.227i)17-s + (0.858 + 1.48i)19-s − 0.901·23-s + 1.36·25-s + (−0.312 − 0.541i)29-s + (0.841 + 1.45i)31-s + (1.06 + 1.10i)35-s + (−0.412 − 0.714i)37-s + (0.187 − 0.325i)41-s + (−0.505 − 0.875i)43-s + (−0.219 + 0.380i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.907 - 0.420i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.907 - 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.107687260\)
\(L(\frac12)\) \(\approx\) \(3.107687260\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-1.83 - 1.90i)T \)
good5 \( 1 - 3.43T + 5T^{2} \)
11 \( 1 - 4.40T + 11T^{2} \)
13 \( 1 + (-1.49 + 2.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.542 - 0.939i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.74 - 6.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.32T + 23T^{2} \)
29 \( 1 + (1.68 + 2.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.68 - 8.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.50 + 4.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.20 + 2.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.31 + 5.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.50 - 2.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.530 + 0.919i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.20 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.71 + 4.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (8.21 - 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.17 - 2.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.60 + 2.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.40 + 11.0i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.703632045484700353560538417745, −8.300493466117218929955195652739, −7.20353262476487161669975419270, −6.18897189475508744871955571968, −5.83049116738626204177101538274, −5.18606920406652230060333735473, −4.04974457651304021733389604359, −3.05597339630604464132415561472, −1.84988681352418355391385890932, −1.41783263367170022092989003273, 1.13071656057004968469692336886, 1.78680017569805867433401339352, 2.88039103155539707988279520312, 4.12885506720165428957801390079, 4.73956056547995894441469840040, 5.70691631587547904732677361106, 6.46000224287867116432395215299, 6.95520682765126724570714021347, 7.911779460084068750123772652169, 8.972459196311254635721848554138

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