Properties

Degree $2$
Conductor $3024$
Sign $0.143 + 0.989i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.59 − 2.75i)5-s + (2.56 + 0.658i)7-s + (−1.59 + 2.75i)11-s + (2.85 − 4.93i)13-s + (0.760 + 1.31i)17-s + (0.641 − 1.11i)19-s + (−1.11 − 1.93i)23-s + (−2.56 + 4.43i)25-s + (3.54 + 6.13i)29-s + 9.42·31-s + (−2.26 − 8.10i)35-s + (0.5 − 0.866i)37-s + (2.80 − 4.85i)41-s + (−3.41 − 5.91i)43-s − 5.82·47-s + ⋯
L(s)  = 1  + (−0.711 − 1.23i)5-s + (0.968 + 0.249i)7-s + (−0.479 + 0.830i)11-s + (0.790 − 1.36i)13-s + (0.184 + 0.319i)17-s + (0.147 − 0.254i)19-s + (−0.233 − 0.404i)23-s + (−0.512 + 0.887i)25-s + (0.657 + 1.13i)29-s + 1.69·31-s + (−0.382 − 1.37i)35-s + (0.0821 − 0.142i)37-s + (0.437 − 0.757i)41-s + (−0.520 − 0.901i)43-s − 0.850·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.143 + 0.989i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.143 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.717386205\)
\(L(\frac12)\) \(\approx\) \(1.717386205\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 0.658i)T \)
good5 \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.59 - 2.75i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.85 + 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.760 - 1.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.641 + 1.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.11 + 1.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.42T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.41 + 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.82T + 47T^{2} \)
53 \( 1 + (1.02 + 1.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + (2.48 + 4.30i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 4.13T + 79T^{2} \)
83 \( 1 + (4.03 + 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.112 - 0.195i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.42 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.398963771863507408159337067067, −8.048876877915964092128459228026, −7.31886398093229333086754313430, −6.14863035713567566293475901717, −5.15999265080213889861850465108, −4.85111144246621631615692822808, −3.98038719368171305050567111948, −2.87857819222402069776386188015, −1.60961972966744631770239461711, −0.63816550227816665519455639918, 1.14540992544256778999823923179, 2.44395589015714153884188980307, 3.31371380634973596045311532899, 4.14504608451351355454099928728, 4.87125187907907945661891461535, 6.14178788032864467449845768212, 6.54501351830025691999425804697, 7.55618814469283132446258279604, 8.021491824590831097790714268462, 8.653097934380902295257409496185

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