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Properties

Label 2-3024-63.4-c1-0-5
Degree $2$
Conductor $3024$
Sign $0.165 - 0.986i$
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  − 2.77·5-s + (−0.855 − 2.50i)7-s − 3.43·11-s + (−0.429 + 0.743i)13-s + (0.405 − 0.701i)17-s + (−0.750 − 1.29i)19-s + 7.64·23-s + 2.68·25-s + (−3.99 − 6.92i)29-s + (−3.60 − 6.24i)31-s + (2.37 + 6.93i)35-s + (0.458 + 0.793i)37-s + (−1.67 + 2.90i)41-s + (−1.20 − 2.08i)43-s + (0.307 − 0.532i)47-s + ⋯
L(s)  = 1  − 1.23·5-s + (−0.323 − 0.946i)7-s − 1.03·11-s + (−0.119 + 0.206i)13-s + (0.0982 − 0.170i)17-s + (−0.172 − 0.298i)19-s + 1.59·23-s + 0.536·25-s + (−0.742 − 1.28i)29-s + (−0.647 − 1.12i)31-s + (0.400 + 1.17i)35-s + (0.0753 + 0.130i)37-s + (−0.261 + 0.453i)41-s + (−0.183 − 0.318i)43-s + (0.0448 − 0.0776i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.165 - 0.986i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.165 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4644934758\)
\(L(\frac12)\) \(\approx\) \(0.4644934758\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.855 + 2.50i)T \)
good5 \( 1 + 2.77T + 5T^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 + (0.429 - 0.743i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.405 + 0.701i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.750 + 1.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.64T + 23T^{2} \)
29 \( 1 + (3.99 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.60 + 6.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.458 - 0.793i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.67 - 2.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.20 + 2.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.307 + 0.532i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.31 - 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.734 + 1.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.71 - 9.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.10 - 14.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + (4.16 - 7.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.75 + 9.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.11 - 8.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.82 - 6.63i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.818791943884103531297732973245, −7.87037539470706175649253599309, −7.51632794094447743155123182430, −6.90040632735538469269498447389, −5.84583282468498298244753800591, −4.83343435894495130590835109870, −4.17205609690504987427583104414, −3.41293692900669311547935035142, −2.48405957207388267759662111628, −0.826271993036940889955331849202, 0.19746052626526666808789607401, 1.89119889171400542794475592544, 3.16125721174603284416038805409, 3.48441425701232192143207168289, 4.94020399945316667893062677987, 5.20642180922355186687848146815, 6.35360363817700056626824462735, 7.18972458715484525443789134452, 7.82885689521935491861301974244, 8.529083604262813317976861315887

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