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Properties

Label 2-3024-252.115-c1-0-2
Degree $2$
Conductor $3024$
Sign $-0.943 - 0.331i$
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  + (2.47 + 1.43i)5-s + (−1.38 − 2.25i)7-s + (−3.57 + 2.06i)11-s + (−3.14 + 1.81i)13-s + (3.36 + 1.94i)17-s + (3.57 + 6.18i)19-s + (−5.18 − 2.99i)23-s + (1.60 + 2.77i)25-s + (2.87 − 4.98i)29-s − 10.4·31-s + (−0.219 − 7.57i)35-s + (−2.02 − 3.50i)37-s + (−2.64 + 1.52i)41-s + (−0.533 − 0.308i)43-s − 1.07·47-s + ⋯
L(s)  = 1  + (1.10 + 0.640i)5-s + (−0.524 − 0.851i)7-s + (−1.07 + 0.622i)11-s + (−0.870 + 0.502i)13-s + (0.815 + 0.470i)17-s + (0.819 + 1.41i)19-s + (−1.08 − 0.623i)23-s + (0.320 + 0.554i)25-s + (0.534 − 0.925i)29-s − 1.88·31-s + (−0.0370 − 1.28i)35-s + (−0.332 − 0.576i)37-s + (−0.412 + 0.238i)41-s + (−0.0814 − 0.0470i)43-s − 0.156·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.943 - 0.331i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6034172424\)
\(L(\frac12)\) \(\approx\) \(0.6034172424\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 + 2.25i)T \)
good5 \( 1 + (-2.47 - 1.43i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.57 - 2.06i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.14 - 1.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.36 - 1.94i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.57 - 6.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.18 + 2.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.87 + 4.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + (2.02 + 3.50i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.64 - 1.52i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.533 + 0.308i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.07T + 47T^{2} \)
53 \( 1 + (-2.65 + 4.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.29T + 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 - 1.58iT - 67T^{2} \)
71 \( 1 + 5.90iT - 71T^{2} \)
73 \( 1 + (6.78 + 3.91i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 4.05iT - 79T^{2} \)
83 \( 1 + (-1.69 + 2.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.80 - 5.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.52 - 1.45i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.407192151793079443220687640547, −8.064115098652476299227695986379, −7.51945637492938188161519543211, −6.84828961831739155164639970187, −5.95169601196577974892770156956, −5.45432025489438420125143425406, −4.33092782460104037981425278539, −3.44433315694355540144251404013, −2.43938338179770410425994437967, −1.65909362688613845613073463349, 0.16787162306022926557526797081, 1.65242852341688034886388411293, 2.71375499632684925411328813932, 3.24745914419537674751748827220, 4.90661091157074818231598224945, 5.45958992044700750606245986107, 5.70676938288313617254584559192, 6.89035165022924103411256560356, 7.66597541039274393824899399616, 8.515611073909905568482648826544

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