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Properties

Label 2-3024-63.41-c1-0-8
Degree $2$
Conductor $3024$
Sign $-0.300 - 0.953i$
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  + (−0.422 + 0.731i)5-s + (−0.327 − 2.62i)7-s + (−0.791 + 0.456i)11-s + (−0.472 − 0.272i)13-s − 4.77·17-s − 3.15i·19-s + (1.39 + 0.804i)23-s + (2.14 + 3.71i)25-s + (−5.56 + 3.21i)29-s + (−1.57 − 0.908i)31-s + (2.05 + 0.869i)35-s + 7.14·37-s + (−2.82 + 4.88i)41-s + (1.84 + 3.19i)43-s + (6.75 + 11.6i)47-s + ⋯
L(s)  = 1  + (−0.188 + 0.327i)5-s + (−0.123 − 0.992i)7-s + (−0.238 + 0.137i)11-s + (−0.131 − 0.0756i)13-s − 1.15·17-s − 0.723i·19-s + (0.290 + 0.167i)23-s + (0.428 + 0.742i)25-s + (−1.03 + 0.596i)29-s + (−0.282 − 0.163i)31-s + (0.348 + 0.146i)35-s + 1.17·37-s + (−0.440 + 0.763i)41-s + (0.281 + 0.487i)43-s + (0.984 + 1.70i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.300 - 0.953i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7677602782\)
\(L(\frac12)\) \(\approx\) \(0.7677602782\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.327 + 2.62i)T \)
good5 \( 1 + (0.422 - 0.731i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.791 - 0.456i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.472 + 0.272i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 + 3.15iT - 19T^{2} \)
23 \( 1 + (-1.39 - 0.804i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.56 - 3.21i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.57 + 0.908i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.14T + 37T^{2} \)
41 \( 1 + (2.82 - 4.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.84 - 3.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.75 - 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.49iT - 53T^{2} \)
59 \( 1 + (-0.279 + 0.483i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.9 - 6.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.06 + 5.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.24iT - 71T^{2} \)
73 \( 1 - 8.87iT - 73T^{2} \)
79 \( 1 + (-5.58 - 9.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.122 - 0.211i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.19T + 89T^{2} \)
97 \( 1 + (12.2 - 7.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.142185279142631249553932429202, −8.021715868989250424336198251460, −7.37284677422721162924025666228, −6.85400332495680238408939833912, −6.02950904306429490880858087605, −4.92642088335483156639447113347, −4.29972016877805081275756955957, −3.37014914990923691329141741214, −2.48698753101372966412329291351, −1.16108941256315795379664842372, 0.25198637917852938124019498145, 1.90218140114224281108008827014, 2.65688773316488991436032630029, 3.79121627774477110788269905731, 4.62438120328062617714605138787, 5.49360103881701642858252196805, 6.11996924632916098164264560654, 6.98764407336480810805569150829, 7.85218744489510354755488850541, 8.654548514631665667807281972549

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