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Properties

Label 2-3024-9.4-c1-0-21
Degree $2$
Conductor $3024$
Sign $0.569 + 0.821i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.724 + 1.25i)5-s + (−0.5 − 0.866i)7-s + (−1 − 1.73i)11-s + (−2.44 + 4.24i)13-s − 2·17-s − 2.55·19-s + (0.5 − 0.866i)23-s + (1.44 + 2.51i)25-s + (−3.44 − 5.97i)29-s + (3 − 5.19i)31-s + 1.44·35-s + 11.7·37-s + (4.89 − 8.48i)41-s + (3.44 + 5.97i)43-s + (−4.89 − 8.48i)47-s + ⋯
L(s)  = 1  + (−0.324 + 0.561i)5-s + (−0.188 − 0.327i)7-s + (−0.301 − 0.522i)11-s + (−0.679 + 1.17i)13-s − 0.485·17-s − 0.585·19-s + (0.104 − 0.180i)23-s + (0.289 + 0.502i)25-s + (−0.640 − 1.10i)29-s + (0.538 − 0.933i)31-s + 0.245·35-s + 1.93·37-s + (0.765 − 1.32i)41-s + (0.526 + 0.911i)43-s + (−0.714 − 1.23i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.121967080\)
\(L(\frac12)\) \(\approx\) \(1.121967080\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.724 - 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.44 + 5.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 + (-4.89 + 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.44 - 5.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.89 + 8.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.27 + 5.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.44 - 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.101T + 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 + (0.949 + 1.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (1.44 + 2.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.621154318678727758978821169438, −7.70850991928327760541600129725, −7.16548997375381068309307579273, −6.39949652874841159680292183085, −5.69670983579081659061004437987, −4.44680875784617455217026858825, −4.03345662324417371017886975532, −2.84876759953526749060520700261, −2.09082958122113165159499681266, −0.42828108338245816140965397411, 0.937259493279072963151882205814, 2.35730703620098233407785598950, 3.09993880699324465818559147216, 4.34534740014173472972634064887, 4.88289543522764685800845048500, 5.73285291554331732886684097131, 6.56204533114860218870645803757, 7.52253698888361949235656196917, 8.020387243165545346022721098859, 8.863961955946735191315102850584

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