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Properties

Label 2-3024-9.4-c1-0-35
Degree $2$
Conductor $3024$
Sign $-0.964 + 0.263i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 − 2.98i)5-s + (−0.5 − 0.866i)7-s + (−1 − 1.73i)11-s + (2.44 − 4.24i)13-s − 2·17-s − 7.44·19-s + (0.5 − 0.866i)23-s + (−3.44 − 5.97i)25-s + (1.44 + 2.51i)29-s + (3 − 5.19i)31-s − 3.44·35-s − 7.79·37-s + (−4.89 + 8.48i)41-s + (−1.44 − 2.51i)43-s + (4.89 + 8.48i)47-s + ⋯
L(s)  = 1  + (0.771 − 1.33i)5-s + (−0.188 − 0.327i)7-s + (−0.301 − 0.522i)11-s + (0.679 − 1.17i)13-s − 0.485·17-s − 1.70·19-s + (0.104 − 0.180i)23-s + (−0.689 − 1.19i)25-s + (0.269 + 0.466i)29-s + (0.538 − 0.933i)31-s − 0.583·35-s − 1.28·37-s + (−0.765 + 1.32i)41-s + (−0.221 − 0.382i)43-s + (0.714 + 1.23i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.964 + 0.263i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.312437827\)
\(L(\frac12)\) \(\approx\) \(1.312437827\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.44 + 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.44 - 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.44 + 2.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.89 - 8.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.10T + 53T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.72 + 9.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.55 - 2.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 + (-3.94 - 6.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.10T + 89T^{2} \)
97 \( 1 + (-3.44 - 5.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.373293775134805274076345813916, −7.999058606039638107417423496361, −6.62155698540137283550764014903, −6.08541880532994953123462513614, −5.27343396883740495907600097923, −4.61494208318840483764078017493, −3.66740409410783654673776884485, −2.53563875895485583929757130106, −1.42631884643770872397490246610, −0.38626747172944161336706018056, 1.88651164777162190135914651856, 2.34188251176951576050834979200, 3.43711485177730441972578419761, 4.31703896368332522902798102218, 5.34991950241466356671536204392, 6.36342228030710423016367168209, 6.62004574339685855107489384451, 7.29103358144423426654618226280, 8.557328132270988421940719362337, 8.952624360315928896598724661350

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