Properties

Label 2-3024-9.7-c1-0-17
Degree $2$
Conductor $3024$
Sign $0.999 + 0.0167i$
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  + (0.590 + 1.02i)5-s + (−0.5 + 0.866i)7-s + (1.85 − 3.20i)11-s + (−0.5 − 0.866i)13-s + 6.94·17-s − 1.94·19-s + (2.80 + 4.85i)23-s + (1.80 − 3.12i)25-s + (0.119 − 0.207i)29-s + (0.830 + 1.43i)31-s − 1.18·35-s − 9.54·37-s + (−5.09 − 8.81i)41-s + (1.11 − 1.92i)43-s + (−2.91 + 5.04i)47-s + ⋯
L(s)  = 1  + (0.264 + 0.457i)5-s + (−0.188 + 0.327i)7-s + (0.558 − 0.967i)11-s + (−0.138 − 0.240i)13-s + 1.68·17-s − 0.445·19-s + (0.584 + 1.01i)23-s + (0.360 − 0.624i)25-s + (0.0222 − 0.0384i)29-s + (0.149 + 0.258i)31-s − 0.199·35-s − 1.56·37-s + (−0.795 − 1.37i)41-s + (0.169 − 0.293i)43-s + (−0.425 + 0.736i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.0167i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.999 + 0.0167i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.042126828\)
\(L(\frac12)\) \(\approx\) \(2.042126828\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.590 - 1.02i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.85 + 3.20i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.94T + 17T^{2} \)
19 \( 1 + 1.94T + 19T^{2} \)
23 \( 1 + (-2.80 - 4.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.119 + 0.207i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.830 - 1.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 + (5.09 + 8.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.11 + 1.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (1.30 + 2.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.80 + 6.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.75 - 3.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.60T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + (-3.68 + 6.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.47 + 6.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.74T + 89T^{2} \)
97 \( 1 + (3.58 - 6.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.712347393601569342656303157269, −8.049595277892579778881938139932, −7.12602812687265258568220262232, −6.46488722617392799469989745451, −5.63197250095149151847708290847, −5.11672599234404282600192874039, −3.63565590071105167766920919343, −3.27646062555408079185391564558, −2.10818628782220362473010350539, −0.849027982994484041888054815674, 0.962313580383277017699458466818, 1.93147905814714645674785952660, 3.14870129668995182039097300607, 4.04094944146420526592716527578, 4.90934243947717320803312909593, 5.53089068503411477886859147252, 6.68196605801969172794052040387, 7.04447269753738587553627980161, 8.051223681439861784462316083872, 8.718607833676697679551049195705

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