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Properties

Label 2-3024-252.115-c1-0-44
Degree $2$
Conductor $3024$
Sign $-0.971 - 0.235i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 2.59i)7-s + (−3 + 1.73i)11-s + (−1.5 + 0.866i)13-s + (−1.5 − 0.866i)17-s + (2.5 + 4.33i)19-s + (3 + 1.73i)23-s + (−2.5 − 4.33i)25-s + (1.5 − 2.59i)29-s − 31-s + (−3.5 − 6.06i)37-s + (1.5 − 0.866i)41-s + (−1.5 − 0.866i)43-s − 9·47-s + (−6.5 − 2.59i)49-s + (−4.5 + 7.79i)53-s + ⋯
L(s)  = 1  + (0.188 − 0.981i)7-s + (−0.904 + 0.522i)11-s + (−0.416 + 0.240i)13-s + (−0.363 − 0.210i)17-s + (0.573 + 0.993i)19-s + (0.625 + 0.361i)23-s + (−0.5 − 0.866i)25-s + (0.278 − 0.482i)29-s − 0.179·31-s + (−0.575 − 0.996i)37-s + (0.234 − 0.135i)41-s + (−0.228 − 0.132i)43-s − 1.31·47-s + (−0.928 − 0.371i)49-s + (−0.618 + 1.07i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.971 - 0.235i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.971 - 0.235i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 15T + 59T^{2} \)
61 \( 1 + 1.73iT - 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.73iT - 79T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.066997328747154801619831775674, −7.62505201097825521250836359400, −6.97488744113172060921543043370, −6.05443060904287548270424202363, −5.10308860182521041724165128218, −4.45633014696605912576247000157, −3.58466230732579036695988515690, −2.52318004478958790371523542127, −1.45517480220919234002557899312, 0, 1.63142536879454911162134474941, 2.76649448282084241038669008163, 3.28183280471140866440377752629, 4.85258244958839139216139905635, 5.10893173834687791087246559606, 6.05433432504374695727134363754, 6.84863104003930034977144189068, 7.75836525291903567664530246783, 8.368993637021589928844114128998

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