Properties

Degree $2$
Conductor $3024$
Sign $0.857 + 0.514i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.340·5-s + (−1.09 − 2.40i)7-s − 0.671·11-s + (1.62 + 2.81i)13-s + (1.10 + 1.90i)17-s + (−0.242 + 0.419i)19-s + 4.18·23-s − 4.88·25-s + (−0.478 + 0.829i)29-s + (1.04 − 1.80i)31-s + (0.373 + 0.818i)35-s + (4.81 − 8.34i)37-s + (3.90 + 6.75i)41-s + (3.66 − 6.34i)43-s + (1.34 + 2.33i)47-s + ⋯
L(s)  = 1  − 0.152·5-s + (−0.414 − 0.909i)7-s − 0.202·11-s + (0.450 + 0.779i)13-s + (0.266 + 0.462i)17-s + (−0.0555 + 0.0961i)19-s + 0.873·23-s − 0.976·25-s + (−0.0889 + 0.154i)29-s + (0.187 − 0.323i)31-s + (0.0631 + 0.138i)35-s + (0.791 − 1.37i)37-s + (0.609 + 1.05i)41-s + (0.558 − 0.967i)43-s + (0.196 + 0.340i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.857 + 0.514i$
Motivic weight: \(1\)
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.857 + 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.604748804\)
\(L(\frac12)\) \(\approx\) \(1.604748804\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 + 2.40i)T \)
good5 \( 1 + 0.340T + 5T^{2} \)
11 \( 1 + 0.671T + 11T^{2} \)
13 \( 1 + (-1.62 - 2.81i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.10 - 1.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.242 - 0.419i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.18T + 23T^{2} \)
29 \( 1 + (0.478 - 0.829i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.04 + 1.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.81 + 8.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.90 - 6.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.66 + 6.34i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.34 - 2.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.12 - 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.47 + 4.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.76 - 3.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.16 + 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.57T + 71T^{2} \)
73 \( 1 + (3.71 + 6.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.00 + 8.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.47 + 4.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.52 + 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.23 + 7.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.759287756383194078295354636658, −7.65469813697669187722729619557, −7.36509713691863148664965308427, −6.32889349869405635535516677552, −5.79185947742186779714654599415, −4.54646335086879531754580342189, −3.99010275081120622945002402629, −3.13501201898629519613844074751, −1.91103153056812731115074669994, −0.68834399567838738419124204041, 0.873152080779529402500948921403, 2.35247238673767741059611807129, 3.06718889847588011362968265500, 3.98588735258881813358056214902, 5.12860643016693840596888256005, 5.66659557594720229279246514885, 6.47042154075259325737161974005, 7.30932194741589997582608629058, 8.164929443483337686604825375387, 8.689170928575039110909631689941

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