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Properties

Label 2-3024-9.4-c1-0-27
Degree $2$
Conductor $3024$
Sign $0.460 + 0.887i$
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  + (2.19 − 3.79i)5-s + (−0.5 − 0.866i)7-s + (2.69 + 4.66i)11-s + (1.27 − 2.20i)13-s + 2.58·17-s + 6.72·19-s + (0.400 − 0.693i)23-s + (−7.09 − 12.2i)25-s + (1.87 + 3.24i)29-s + (1.69 − 2.93i)31-s − 4.38·35-s + 4.38·37-s + (−3.19 + 5.53i)41-s + (−0.381 − 0.661i)43-s + (4.13 + 7.16i)47-s + ⋯
L(s)  = 1  + (0.979 − 1.69i)5-s + (−0.188 − 0.327i)7-s + (0.811 + 1.40i)11-s + (0.352 − 0.610i)13-s + 0.625·17-s + 1.54·19-s + (0.0834 − 0.144i)23-s + (−1.41 − 2.45i)25-s + (0.347 + 0.601i)29-s + (0.304 − 0.527i)31-s − 0.740·35-s + 0.720·37-s + (−0.499 + 0.864i)41-s + (−0.0581 − 0.100i)43-s + (0.603 + 1.04i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.460 + 0.887i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.582246371\)
\(L(\frac12)\) \(\approx\) \(2.582246371\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 3.79i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.69 - 4.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.27 + 2.20i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 - 6.72T + 19T^{2} \)
23 \( 1 + (-0.400 + 0.693i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.87 - 3.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.69 + 2.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.38T + 37T^{2} \)
41 \( 1 + (3.19 - 5.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.381 + 0.661i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.13 - 7.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.94T + 53T^{2} \)
59 \( 1 + (2.78 - 4.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.14 - 7.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.946 - 1.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.34T + 71T^{2} \)
73 \( 1 + 8.65T + 73T^{2} \)
79 \( 1 + (6.64 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.86 + 3.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.99T + 89T^{2} \)
97 \( 1 + (1.48 + 2.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.748105372237664210307822028420, −7.889332430323087578459293682609, −7.16857463480351103806418860236, −6.11809112037735926748129888789, −5.49878311007369358150457725761, −4.73383258220406374972579069592, −4.12078682392186730616660447835, −2.82596076244788986177645948345, −1.51082273483431850333379143014, −1.00561266282661451801774813100, 1.22917870345209038998300422514, 2.41874910870923766428352748612, 3.24169903761344562155933634829, 3.73742864471056764724299899600, 5.39336536835169195185313272003, 5.88225470357169157586100050011, 6.58318129907186358387373421126, 7.09357978942519994596227277818, 8.079418641004746843305459692597, 9.033178068697149124609330862746

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