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Properties

Label 2-3024-9.7-c1-0-31
Degree $2$
Conductor $3024$
Sign $-0.989 + 0.145i$
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  + (−1.79 − 3.10i)5-s + (−0.5 + 0.866i)7-s + (1.40 − 2.43i)11-s + (−0.5 − 0.866i)13-s + 4.11·17-s + 0.888·19-s + (−2.93 − 5.08i)23-s + (−3.93 + 6.82i)25-s + (−0.849 + 1.47i)29-s + (−3.49 − 6.05i)31-s + 3.58·35-s + 4.76·37-s + (−2.70 − 4.68i)41-s + (2.60 − 4.51i)43-s + (1.33 − 2.30i)47-s + ⋯
L(s)  = 1  + (−0.802 − 1.38i)5-s + (−0.188 + 0.327i)7-s + (0.423 − 0.733i)11-s + (−0.138 − 0.240i)13-s + 0.997·17-s + 0.203·19-s + (−0.612 − 1.06i)23-s + (−0.787 + 1.36i)25-s + (−0.157 + 0.273i)29-s + (−0.627 − 1.08i)31-s + 0.606·35-s + 0.783·37-s + (−0.422 − 0.731i)41-s + (0.397 − 0.688i)43-s + (0.194 − 0.336i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.989 + 0.145i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.989 + 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8595215494\)
\(L(\frac12)\) \(\approx\) \(0.8595215494\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 + 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.40 + 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.11T + 17T^{2} \)
19 \( 1 - 0.888T + 19T^{2} \)
23 \( 1 + (2.93 + 5.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.849 - 1.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.49 + 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 + (2.70 + 4.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.60 + 4.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.123T + 53T^{2} \)
59 \( 1 + (-4.43 - 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.15 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + (3.54 - 6.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.05 + 3.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 + (3.66 - 6.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.422112081870937423916108677367, −7.82205844782544671210862908991, −6.97941467045287398692897154262, −5.74827835080202398171144647915, −5.46376350110098198065087812117, −4.29319534976527947596865612974, −3.81882291575814922938370235794, −2.66740827258942787969007912191, −1.25889491061855332483886152606, −0.29483179020966090786849656840, 1.50737773355295422680364263169, 2.78031442007244614784697978302, 3.53395986844692356720238544089, 4.15292042216568529658043095214, 5.24641079329610588286036058449, 6.30865017544807050206192954765, 6.87596112935053817217286951365, 7.63421689447499730626294056144, 7.927924480122346704668441881768, 9.266539907078583522495126430956

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