Properties

Label 2-3024-63.41-c1-0-13
Degree $2$
Conductor $3024$
Sign $-0.901 - 0.433i$
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  + (−2.09 + 3.62i)5-s + (2.64 − 0.0532i)7-s + (−1.23 + 0.711i)11-s + (0.850 + 0.491i)13-s + 0.370·17-s + 4.97i·19-s + (4.98 + 2.87i)23-s + (−6.26 − 10.8i)25-s + (−7.31 + 4.22i)29-s + (6.28 + 3.62i)31-s + (−5.34 + 9.70i)35-s + 3.46·37-s + (−1.06 + 1.85i)41-s + (−3.00 − 5.21i)43-s + (4.13 + 7.16i)47-s + ⋯
L(s)  = 1  + (−0.936 + 1.62i)5-s + (0.999 − 0.0201i)7-s + (−0.371 + 0.214i)11-s + (0.235 + 0.136i)13-s + 0.0899·17-s + 1.14i·19-s + (1.04 + 0.600i)23-s + (−1.25 − 2.17i)25-s + (−1.35 + 0.784i)29-s + (1.12 + 0.651i)31-s + (−0.903 + 1.64i)35-s + 0.569·37-s + (−0.167 + 0.289i)41-s + (−0.458 − 0.794i)43-s + (0.603 + 1.04i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.901 - 0.433i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.901 - 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.257751890\)
\(L(\frac12)\) \(\approx\) \(1.257751890\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.0532i)T \)
good5 \( 1 + (2.09 - 3.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.23 - 0.711i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.850 - 0.491i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.370T + 17T^{2} \)
19 \( 1 - 4.97iT - 19T^{2} \)
23 \( 1 + (-4.98 - 2.87i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.31 - 4.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.28 - 3.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.46T + 37T^{2} \)
41 \( 1 + (1.06 - 1.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.00 + 5.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.13 - 7.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.97iT - 53T^{2} \)
59 \( 1 + (-2.27 + 3.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.50 - 3.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.03 - 8.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 9.52iT - 73T^{2} \)
79 \( 1 + (4.25 + 7.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.972 + 1.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.80T + 89T^{2} \)
97 \( 1 + (-3.34 + 1.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.887815786855257840838589819021, −8.053391865786299173233066910648, −7.53123369046303974683533691248, −7.01132678891147100038374410492, −6.10903258093689001304452242439, −5.19378732684056262913143780481, −4.20824811027408156131867683068, −3.46211491778554364075460823891, −2.67112750112691655070852446345, −1.50302582506875788749245059018, 0.42916984662637562173461469756, 1.31187155333133311222775405115, 2.61452454765311211121627089108, 3.91079159010438384702860351731, 4.59817817895417855645689376810, 5.08614743658981699071858011978, 5.89080963941075187548275168252, 7.17784055049646987281608436243, 7.85085031375370219071637987139, 8.383206373350103326005944202631

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