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Properties

Label 2-3024-63.41-c1-0-1
Degree $2$
Conductor $3024$
Sign $-0.671 + 0.741i$
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  + (−1.16 + 2.01i)5-s + (1.39 + 2.24i)7-s + (−2.17 + 1.25i)11-s + (−0.244 − 0.141i)13-s − 2.01·17-s − 2.93i·19-s + (−4.32 − 2.49i)23-s + (−0.199 − 0.345i)25-s + (−4.02 + 2.32i)29-s + (−8.91 − 5.14i)31-s + (−6.14 + 0.191i)35-s + 3.93·37-s + (−3.44 + 5.97i)41-s + (5.66 + 9.80i)43-s + (−1.84 − 3.19i)47-s + ⋯
L(s)  = 1  + (−0.519 + 0.899i)5-s + (0.526 + 0.850i)7-s + (−0.656 + 0.379i)11-s + (−0.0678 − 0.0391i)13-s − 0.487·17-s − 0.674i·19-s + (−0.901 − 0.520i)23-s + (−0.0398 − 0.0690i)25-s + (−0.747 + 0.431i)29-s + (−1.60 − 0.924i)31-s + (−1.03 + 0.0324i)35-s + 0.646·37-s + (−0.538 + 0.932i)41-s + (0.863 + 1.49i)43-s + (−0.268 − 0.465i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.671 + 0.741i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.671 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08475914135\)
\(L(\frac12)\) \(\approx\) \(0.08475914135\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 - 2.24i)T \)
good5 \( 1 + (1.16 - 2.01i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.17 - 1.25i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.244 + 0.141i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.01T + 17T^{2} \)
19 \( 1 + 2.93iT - 19T^{2} \)
23 \( 1 + (4.32 + 2.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.02 - 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.91 + 5.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.93T + 37T^{2} \)
41 \( 1 + (3.44 - 5.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.66 - 9.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.84 + 3.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.845iT - 53T^{2} \)
59 \( 1 + (-7.27 + 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.59 + 5.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.59 + 9.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.67iT - 71T^{2} \)
73 \( 1 - 6.37iT - 73T^{2} \)
79 \( 1 + (3.71 + 6.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.73 + 8.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 + (4.49 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.246527526394049194294681094329, −8.261597348860370229161304815859, −7.76187134609313692605592727191, −7.00747912040826045753873561530, −6.22140395965341913634483420209, −5.34876522236377023260402284104, −4.59144782006606236364559455708, −3.59672443502422798745805920212, −2.63631663242982292894608460740, −1.94571891613554889591580234641, 0.02691633380430529516233524327, 1.21531338100171033379971328915, 2.32798792097236364241931426192, 3.87333377264357390415507125374, 4.06097711962315567229680184811, 5.25768162707541652138120446020, 5.66214022962417666945426179856, 7.01679393804431417804955448066, 7.52395957991707325823479214081, 8.333396911916598830996379964122

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