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Properties

Label 2-3024-252.31-c1-0-3
Degree $2$
Conductor $3024$
Sign $-0.936 - 0.350i$
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  + 4.18i·5-s + (2.62 + 0.358i)7-s − 1.73i·11-s + (−0.621 − 0.358i)13-s + (−5.74 − 3.31i)17-s + (−0.5 − 0.866i)19-s + 7.64i·23-s − 12.4·25-s + (3.62 + 6.27i)29-s + (2 + 3.46i)31-s + (−1.5 + 10.9i)35-s + (−2.62 − 4.54i)37-s + (−0.257 − 0.148i)41-s + (−2.74 + 1.58i)43-s + (−4.24 + 7.34i)47-s + ⋯
L(s)  = 1  + 1.87i·5-s + (0.990 + 0.135i)7-s − 0.522i·11-s + (−0.172 − 0.0994i)13-s + (−1.39 − 0.804i)17-s + (−0.114 − 0.198i)19-s + 1.59i·23-s − 2.49·25-s + (0.672 + 1.16i)29-s + (0.359 + 0.622i)31-s + (−0.253 + 1.85i)35-s + (−0.430 − 0.746i)37-s + (−0.0401 − 0.0232i)41-s + (−0.418 + 0.241i)43-s + (−0.618 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.936 - 0.350i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.235590487\)
\(L(\frac12)\) \(\approx\) \(1.235590487\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.358i)T \)
good5 \( 1 - 4.18iT - 5T^{2} \)
11 \( 1 + 1.73iT - 11T^{2} \)
13 \( 1 + (0.621 + 0.358i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.74 + 3.31i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.64iT - 23T^{2} \)
29 \( 1 + (-3.62 - 6.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.257 + 0.148i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.74 - 1.58i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.24 - 7.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.62 - 6.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.33iT - 71T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.2 + 5.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.74 - 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (11.2 - 6.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.74 - 1.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.026807741807784846730604141588, −8.227090444865671350473414704782, −7.32896815073119214660945112294, −6.97955750399475917887755200599, −6.13142492218406575536031215104, −5.27761375223668935758372884113, −4.35211659695732833136407483731, −3.25661582962966658840388089558, −2.67989482759708372929524221238, −1.64070874087439401346639489520, 0.36681513316924999334803260428, 1.57881282277603315090241755437, 2.27133087466865659976904143963, 4.11271409832530528430017412849, 4.50938653974577760742013822305, 5.03374087925913651418583434321, 6.01950424483915940551207770108, 6.87301580554985984793963510404, 8.042558710416670001024069486937, 8.399493345121246946368216530228

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