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Properties

Label 2-3024-9.7-c1-0-11
Degree $2$
Conductor $3024$
Sign $0.173 - 0.984i$
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 + 1.73i)5-s + (0.5 − 0.866i)7-s + (1.5 − 2.59i)11-s + (3 + 5.19i)13-s − 7·17-s − 19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + (−2 + 3.46i)29-s + (5 + 8.66i)31-s + 1.99·35-s − 6·37-s + (3.5 + 6.06i)41-s + (5.5 − 9.52i)43-s + (−4 + 6.92i)47-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.188 − 0.327i)7-s + (0.452 − 0.783i)11-s + (0.832 + 1.44i)13-s − 1.69·17-s − 0.229·19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + (−0.371 + 0.643i)29-s + (0.898 + 1.55i)31-s + 0.338·35-s − 0.986·37-s + (0.546 + 0.946i)41-s + (0.838 − 1.45i)43-s + (−0.583 + 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.881744065\)
\(L(\frac12)\) \(\approx\) \(1.881744065\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-3.5 - 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.908293110457171107099946544314, −8.336712215107773586376515877181, −7.05008913603601239231417484614, −6.67389264911134285087102454765, −6.12950406590261967610152802013, −4.97867657599210088956583438386, −4.11992831882643716261625713497, −3.33514538041801455050910422042, −2.25866294302774042266442266518, −1.29535007741208165161316550300, 0.61249580301976883138315263209, 1.83607417044362600874233249935, 2.68873000176216713956614496240, 4.00387616593165110208195189123, 4.66196834961513079603373925205, 5.50782758879027996837679472859, 6.19711627725221995093528450979, 6.98023831082236461962561096866, 8.033888584706591706501465227680, 8.556445886798652543988696304271

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