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Properties

Label 2-3024-63.4-c1-0-21
Degree $2$
Conductor $3024$
Sign $0.376 + 0.926i$
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  − 1.42·5-s + (−2.21 + 1.44i)7-s − 4.93·11-s + (−1.37 + 2.38i)13-s + (−0.559 + 0.969i)17-s + (2.00 + 3.47i)19-s + 5.43·23-s − 2.96·25-s + (−3.40 − 5.89i)29-s + (1.25 + 2.17i)31-s + (3.15 − 2.06i)35-s + (0.709 + 1.22i)37-s + (−0.124 + 0.215i)41-s + (0.498 + 0.863i)43-s + (4.73 − 8.20i)47-s + ⋯
L(s)  = 1  − 0.637·5-s + (−0.837 + 0.546i)7-s − 1.48·11-s + (−0.381 + 0.661i)13-s + (−0.135 + 0.235i)17-s + (0.460 + 0.797i)19-s + 1.13·23-s − 0.593·25-s + (−0.632 − 1.09i)29-s + (0.225 + 0.389i)31-s + (0.533 − 0.348i)35-s + (0.116 + 0.202i)37-s + (−0.0194 + 0.0336i)41-s + (0.0759 + 0.131i)43-s + (0.691 − 1.19i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5798637952\)
\(L(\frac12)\) \(\approx\) \(0.5798637952\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 1.44i)T \)
good5 \( 1 + 1.42T + 5T^{2} \)
11 \( 1 + 4.93T + 11T^{2} \)
13 \( 1 + (1.37 - 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.559 - 0.969i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.00 - 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.43T + 23T^{2} \)
29 \( 1 + (3.40 + 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.709 - 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.124 - 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.498 - 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.73 + 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.410 + 0.710i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.29 - 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0376 - 0.0651i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.29 + 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 + 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.922 - 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.23 + 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.76 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.575324800042048046449350011597, −7.73516287772555968448107824043, −7.26127940164729700951633619123, −6.25067492888863052847810233278, −5.54579141868494534742958296645, −4.73095314820213016763064612387, −3.73346372245176793379497153403, −2.93425409655510543051519682823, −2.03588313008645051183641630455, −0.25057801309549442970415935160, 0.78399588994178851340533291397, 2.59496091561758806064677053791, 3.14095848856194203005368801605, 4.12454602842669963268449133897, 5.07120386885960302179194827555, 5.67178137345903714056061674345, 6.85758986466414103569116662989, 7.39829676182349261569439260971, 7.914365677802398839025876313938, 8.879215808779495166725730485766

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