Properties

Degree $2$
Conductor $3024$
Sign $0.245 - 0.969i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.84 − 3.19i)5-s + (−0.926 + 2.47i)7-s + (0.446 − 0.772i)11-s + (0.598 − 1.03i)13-s + (0.124 + 0.216i)17-s + (−1.40 + 2.43i)19-s + (−1.23 − 2.14i)23-s + (−4.31 + 7.47i)25-s + (−2.07 − 3.58i)29-s − 3.58·31-s + (9.63 − 1.61i)35-s + (−2.36 + 4.09i)37-s + (2.39 − 4.14i)41-s + (4.98 + 8.64i)43-s − 10.1·47-s + ⋯
L(s)  = 1  + (−0.825 − 1.43i)5-s + (−0.350 + 0.936i)7-s + (0.134 − 0.233i)11-s + (0.165 − 0.287i)13-s + (0.0303 + 0.0525i)17-s + (−0.322 + 0.557i)19-s + (−0.258 − 0.447i)23-s + (−0.863 + 1.49i)25-s + (−0.384 − 0.666i)29-s − 0.643·31-s + (1.62 − 0.272i)35-s + (−0.388 + 0.673i)37-s + (0.373 − 0.646i)41-s + (0.760 + 1.31i)43-s − 1.48·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.245 - 0.969i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.245 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7176553889\)
\(L(\frac12)\) \(\approx\) \(0.7176553889\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.926 - 2.47i)T \)
good5 \( 1 + (1.84 + 3.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.446 + 0.772i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.598 + 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.124 - 0.216i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.40 - 2.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 + 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.07 + 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.39 + 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.98 - 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + (-4.94 - 8.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.81T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 1.02T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + (0.915 + 1.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.79T + 79T^{2} \)
83 \( 1 + (-6.16 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.20 + 2.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.52 - 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.840199766971154523314833784815, −8.195877463681723321051523671650, −7.69947976855418415096858025536, −6.47712171211477941992667303674, −5.73729040347818591417858851792, −5.04837336112228676984920882970, −4.19611356774658978689839731986, −3.44797605777402094739255526573, −2.22830315852088177247707373443, −1.00140165901038985209169085773, 0.26469765772822829096414647764, 1.94359756965952018895065764651, 3.12092835655001752145802274712, 3.71224734505670385440837143364, 4.38913000384277128956657002847, 5.60055421637929830691009951869, 6.67321591995150158319799632361, 7.02204299275481413211275039587, 7.56548877739585900849045706294, 8.440669710089679993772239937607

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