Properties

Label 2-3024-21.20-c1-0-58
Degree $2$
Conductor $3024$
Sign $-0.560 + 0.828i$
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  + 1.22·5-s + (1.48 − 2.19i)7-s − 3.31i·11-s + 1.60i·13-s − 6.11·17-s + 1.35i·19-s − 1.11i·23-s − 3.50·25-s − 9.39i·29-s − 7.00i·31-s + (1.81 − 2.67i)35-s − 11.7·37-s + 5.86·41-s + 8.58·43-s − 3.15·47-s + ⋯
L(s)  = 1  + 0.546·5-s + (0.560 − 0.828i)7-s − 0.998i·11-s + 0.443i·13-s − 1.48·17-s + 0.310i·19-s − 0.232i·23-s − 0.701·25-s − 1.74i·29-s − 1.25i·31-s + (0.306 − 0.452i)35-s − 1.92·37-s + 0.916·41-s + 1.30·43-s − 0.460·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.560 + 0.828i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.560 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.375794013\)
\(L(\frac12)\) \(\approx\) \(1.375794013\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 2.19i)T \)
good5 \( 1 - 1.22T + 5T^{2} \)
11 \( 1 + 3.31iT - 11T^{2} \)
13 \( 1 - 1.60iT - 13T^{2} \)
17 \( 1 + 6.11T + 17T^{2} \)
19 \( 1 - 1.35iT - 19T^{2} \)
23 \( 1 + 1.11iT - 23T^{2} \)
29 \( 1 + 9.39iT - 29T^{2} \)
31 \( 1 + 7.00iT - 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 - 5.86T + 41T^{2} \)
43 \( 1 - 8.58T + 43T^{2} \)
47 \( 1 + 3.15T + 47T^{2} \)
53 \( 1 + 0.539iT - 53T^{2} \)
59 \( 1 + 6.87T + 59T^{2} \)
61 \( 1 - 7.40iT - 61T^{2} \)
67 \( 1 - 5.74T + 67T^{2} \)
71 \( 1 + 4.81iT - 71T^{2} \)
73 \( 1 - 14.5iT - 73T^{2} \)
79 \( 1 + 1.15T + 79T^{2} \)
83 \( 1 + 7.27T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 - 4.04iT - 97T^{2} \)
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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

−8.436810744243544556171376368427, −7.78152405954991195026675506772, −6.92441122351264540332559484486, −6.15331415366142599055071659303, −5.52911315957681477825364905750, −4.33726027947422667542983436654, −3.96909373900486992710478852034, −2.58435211429973895294413631880, −1.73438293335475765801081755266, −0.39663839599147124584114292535, 1.60880737412305094886638750212, 2.24366938596781140335327343297, 3.28644382746340302585667825755, 4.55560893409758191712845918874, 5.09232326422297579361719897057, 5.86746570707250423947371205506, 6.78292488348428860247854214832, 7.37904110688741637764576275388, 8.408017644957258683740449496962, 8.994016450912477994969091429910

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