Properties

Degree $2$
Conductor $3024$
Sign $-0.395 - 0.918i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.83·5-s + (−2.45 − 0.997i)7-s − 3.09·11-s + (2.40 − 4.16i)13-s + (−1.87 + 3.24i)17-s + (2.71 + 4.70i)19-s − 7.95·23-s − 1.62·25-s + (0.325 + 0.563i)29-s + (0.518 + 0.897i)31-s + (−4.50 − 1.83i)35-s + (0.873 + 1.51i)37-s + (−2.52 + 4.36i)41-s + (6.09 + 10.5i)43-s + (2.30 − 3.99i)47-s + ⋯
L(s)  = 1  + 0.821·5-s + (−0.926 − 0.376i)7-s − 0.933·11-s + (0.666 − 1.15i)13-s + (−0.453 + 0.786i)17-s + (0.622 + 1.07i)19-s − 1.65·23-s − 0.325·25-s + (0.0604 + 0.104i)29-s + (0.0930 + 0.161i)31-s + (−0.760 − 0.309i)35-s + (0.143 + 0.248i)37-s + (−0.393 + 0.682i)41-s + (0.929 + 1.61i)43-s + (0.336 − 0.582i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.395 - 0.918i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8343262141\)
\(L(\frac12)\) \(\approx\) \(0.8343262141\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.45 + 0.997i)T \)
good5 \( 1 - 1.83T + 5T^{2} \)
11 \( 1 + 3.09T + 11T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.87 - 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.71 - 4.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.95T + 23T^{2} \)
29 \( 1 + (-0.325 - 0.563i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.518 - 0.897i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.873 - 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.52 - 4.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.09 - 10.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.30 + 3.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.55 - 7.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.89 - 5.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.40 + 4.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.23 + 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.00T + 71T^{2} \)
73 \( 1 + (1.81 - 3.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.17 - 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.83 - 6.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.76 - 9.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.04 + 1.80i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.039906880948420444457527327347, −7.949737380158847231486000949607, −7.77298229759345597447095902195, −6.34814156036715893339403414719, −6.04711690644643803407118003527, −5.36775816274922003347979159496, −4.14237150261114739195101334357, −3.34200521814270611973832972252, −2.45879445529633607670805760846, −1.28619963059832355021404710975, 0.25145267583119973054266731826, 1.97889561161249094776526130991, 2.58694172417336349271873929765, 3.66895687277347090034866339972, 4.63918057717786583464932768326, 5.62855713754194284105340588988, 6.09979534543319427648274084887, 6.92878387711731052665489479974, 7.61022951653300623088097401420, 8.850098449499988529886859794558

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