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Properties

Label 2-3024-9.7-c1-0-33
Degree $2$
Conductor $3024$
Sign $-0.939 - 0.342i$
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  + (−1.43 − 2.49i)5-s + (0.5 − 0.866i)7-s + (0.592 − 1.02i)11-s + (2.37 + 4.12i)13-s − 5.41·17-s − 1.10·19-s + (−2.95 − 5.11i)23-s + (−1.64 + 2.84i)25-s + (2.49 − 4.31i)29-s + (−2.78 − 4.82i)31-s − 2.87·35-s − 2.42·37-s + (3.81 + 6.61i)41-s + (−3.78 + 6.55i)43-s + (−0.141 + 0.245i)47-s + ⋯
L(s)  = 1  + (−0.643 − 1.11i)5-s + (0.188 − 0.327i)7-s + (0.178 − 0.309i)11-s + (0.659 + 1.14i)13-s − 1.31·17-s − 0.253·19-s + (−0.615 − 1.06i)23-s + (−0.329 + 0.569i)25-s + (0.462 − 0.801i)29-s + (−0.500 − 0.866i)31-s − 0.486·35-s − 0.398·37-s + (0.596 + 1.03i)41-s + (−0.577 + 1.00i)43-s + (−0.0206 + 0.0357i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3482131910\)
\(L(\frac12)\) \(\approx\) \(0.3482131910\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 + 2.49i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.592 + 1.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.37 - 4.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
19 \( 1 + 1.10T + 19T^{2} \)
23 \( 1 + (2.95 + 5.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.49 + 4.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.78 + 4.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.42T + 37T^{2} \)
41 \( 1 + (-3.81 - 6.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.78 - 6.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.141 - 0.245i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.22T + 53T^{2} \)
59 \( 1 + (5.86 + 10.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.992 - 1.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.76 + 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.11T + 71T^{2} \)
73 \( 1 + 0.327T + 73T^{2} \)
79 \( 1 + (5.24 - 9.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.62 - 6.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (9.04 - 15.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.287247487830005555469297426013, −7.83959162584428065588573345146, −6.60104727161628189956416380468, −6.25312783039569531100152040091, −4.93843990076790438857183774191, −4.32089722929385207954572669375, −3.89260207664675937446732845855, −2.40454200829383873960434090745, −1.31685595156219201683337148838, −0.11071984697116168035938306492, 1.66643195194677764484667680180, 2.80393865755141530527180812304, 3.50832140759804226001176195303, 4.33036830904118725642975190966, 5.42951385907103484668608827893, 6.12957815529499766275621692816, 7.13876109867756107045260432569, 7.36887769591069400975652403700, 8.532478712967348259263753051063, 8.851074434014765538054063135929

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