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Properties

Label 2-3024-63.4-c1-0-23
Degree $2$
Conductor $3024$
Sign $0.384 + 0.923i$
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  − 3.69·5-s + (1.40 − 2.24i)7-s − 1.47·11-s + (−1.34 + 2.33i)13-s + (−3.28 + 5.69i)17-s + (0.444 + 0.769i)19-s + 6.28·23-s + 8.68·25-s + (−1.25 − 2.17i)29-s + (3.40 + 5.89i)31-s + (−5.19 + 8.29i)35-s + (−1.38 − 2.40i)37-s + (2.05 − 3.56i)41-s + (−0.00618 − 0.0107i)43-s + (3.49 − 6.05i)47-s + ⋯
L(s)  = 1  − 1.65·5-s + (0.531 − 0.847i)7-s − 0.445·11-s + (−0.374 + 0.648i)13-s + (−0.797 + 1.38i)17-s + (0.101 + 0.176i)19-s + 1.31·23-s + 1.73·25-s + (−0.233 − 0.403i)29-s + (0.611 + 1.05i)31-s + (−0.878 + 1.40i)35-s + (−0.228 − 0.395i)37-s + (0.321 − 0.556i)41-s + (−0.000943 − 0.00163i)43-s + (0.509 − 0.882i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.384 + 0.923i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.384 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9112168142\)
\(L(\frac12)\) \(\approx\) \(0.9112168142\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 2.24i)T \)
good5 \( 1 + 3.69T + 5T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + (1.34 - 2.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.28 - 5.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.444 - 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.28T + 23T^{2} \)
29 \( 1 + (1.25 + 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.40 - 5.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.38 + 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.05 + 3.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.00618 + 0.0107i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.49 + 6.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.60 + 2.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.45 + 5.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.86 + 4.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.73 + 8.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (6.03 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.72 + 9.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.23 - 3.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.43 - 7.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.58 + 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.396827055707767924454575005138, −7.87911650093288332425844197955, −7.12439388104526598059378564580, −6.66862331723558664414112443993, −5.28998434395987895784077409745, −4.45253581731526914383886076774, −4.00412831026464260914192353779, −3.13431535518149902949098713105, −1.75060424794875444397524649592, −0.40316216015751896798635060732, 0.832304183798444563607617541904, 2.60184900123411850407772288709, 3.07012230562984924755884628626, 4.37492768319048147274894619765, 4.83087433698599963435198240723, 5.65363887294831478649261758581, 6.84846782104924953839602657859, 7.54384488125181358179696970679, 7.967643201486467469165449611097, 8.839083785682893340318130139530

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