Properties

Degree $2$
Conductor $3024$
Sign $-0.984 - 0.174i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.68·5-s + (−0.960 + 2.46i)7-s + 1.24·11-s + (1.96 + 3.39i)13-s + (1.62 + 2.81i)17-s + (−2.36 + 4.09i)19-s − 0.398·23-s − 2.16·25-s + (3.19 − 5.54i)29-s + (−0.289 + 0.500i)31-s + (1.61 − 4.14i)35-s + (2.72 − 4.71i)37-s + (−4.20 − 7.27i)41-s + (−2.46 + 4.26i)43-s + (0.212 + 0.368i)47-s + ⋯
L(s)  = 1  − 0.752·5-s + (−0.362 + 0.931i)7-s + 0.375·11-s + (0.543 + 0.941i)13-s + (0.394 + 0.683i)17-s + (−0.541 + 0.938i)19-s − 0.0830·23-s − 0.433·25-s + (0.594 − 1.02i)29-s + (−0.0519 + 0.0899i)31-s + (0.273 − 0.701i)35-s + (0.447 − 0.774i)37-s + (−0.656 − 1.13i)41-s + (−0.375 + 0.650i)43-s + (0.0310 + 0.0537i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.984 - 0.174i$
Motivic weight: \(1\)
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.984 - 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6293728482\)
\(L(\frac12)\) \(\approx\) \(0.6293728482\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.960 - 2.46i)T \)
good5 \( 1 + 1.68T + 5T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 + (-1.96 - 3.39i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.398T + 23T^{2} \)
29 \( 1 + (-3.19 + 5.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.289 - 0.500i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.20 + 7.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.46 - 4.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.212 - 0.368i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.466 - 0.807i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.02 - 5.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.10 + 8.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.70 - 8.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.76 + 4.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.03 - 13.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.03 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.993710238421737348907631508254, −8.306158217100995172771725662239, −7.77449288790412925574298143878, −6.64570562234630116980284922372, −6.15572316524848333874594157682, −5.34804815478913862356258663400, −4.07485645804353533198598218333, −3.80057413881095369067822902644, −2.54116297320809723794689597217, −1.52998797709951417165795249310, 0.21163708877284021972454733321, 1.27585291756561469285307149003, 2.91864710666365911582084406015, 3.52750551292125262268147010307, 4.40056322053705449014442824685, 5.14137875520472747105361600539, 6.29335469822051325031570292733, 6.87861626788861099611226284123, 7.67779423282842453187967434207, 8.222031289034292223366198654505

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