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Properties

Label 2-3024-63.4-c1-0-2
Degree $2$
Conductor $3024$
Sign $-0.609 - 0.792i$
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  − 0.593·5-s + (0.0665 − 2.64i)7-s − 0.593·11-s + (−1.25 + 2.17i)13-s + (−1.46 + 2.52i)17-s + (−2.69 − 4.66i)19-s + 4.46·23-s − 4.64·25-s + (3.09 + 5.36i)29-s + (−3.93 − 6.81i)31-s + (−0.0394 + 1.56i)35-s + (0.5 + 0.866i)37-s + (0.136 − 0.236i)41-s + (5.58 + 9.66i)43-s + (−6.08 + 10.5i)47-s + ⋯
L(s)  = 1  − 0.265·5-s + (0.0251 − 0.999i)7-s − 0.178·11-s + (−0.348 + 0.603i)13-s + (−0.354 + 0.613i)17-s + (−0.617 − 1.06i)19-s + 0.930·23-s − 0.929·25-s + (0.575 + 0.996i)29-s + (−0.706 − 1.22i)31-s + (−0.00667 + 0.265i)35-s + (0.0821 + 0.142i)37-s + (0.0213 − 0.0369i)41-s + (0.851 + 1.47i)43-s + (−0.887 + 1.53i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.609 - 0.792i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4368726735\)
\(L(\frac12)\) \(\approx\) \(0.4368726735\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0665 + 2.64i)T \)
good5 \( 1 + 0.593T + 5T^{2} \)
11 \( 1 + 0.593T + 11T^{2} \)
13 \( 1 + (1.25 - 2.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.46 - 2.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.46T + 23T^{2} \)
29 \( 1 + (-3.09 - 5.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.93 + 6.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.58 - 9.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.08 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.02 - 6.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.32 + 7.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.32 + 5.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.956 + 1.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (-3.95 + 6.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.62 - 8.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.85 - 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.21 - 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.86 - 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.153083695996690936501550947041, −8.046921770047462294595826769028, −7.56834581849613335742772788736, −6.72810743527269302460310904144, −6.16758505230805104151677337661, −4.85402549008025551271074926955, −4.41357105382514782688308722520, −3.50563631466595771172317581845, −2.45532829379490195336438340564, −1.24972310711809231055193318103, 0.13811237703275275945094273193, 1.81387882155805111513604093031, 2.71364309057279877397518075675, 3.60193562419283687915475595589, 4.65314004553147250613803840458, 5.43937107592793750848811380115, 6.04614469582313632061930801385, 7.03702935186162363603925086759, 7.73318392628776622297481318340, 8.578521162049119006075566360218

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