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Properties

Label 2-3024-252.115-c1-0-39
Degree $2$
Conductor $3024$
Sign $-0.462 + 0.886i$
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  + (2.08 + 1.20i)5-s + (−2.53 + 0.763i)7-s + (−2.81 + 1.62i)11-s + (4.35 − 2.51i)13-s + (−0.795 − 0.459i)17-s + (−3.22 − 5.57i)19-s + (−5.31 − 3.06i)23-s + (0.399 + 0.691i)25-s + (−1.22 + 2.12i)29-s − 3.21·31-s + (−6.20 − 1.45i)35-s + (−4.08 − 7.07i)37-s + (−10.3 + 5.99i)41-s + (10.2 + 5.89i)43-s + 4.84·47-s + ⋯
L(s)  = 1  + (0.932 + 0.538i)5-s + (−0.957 + 0.288i)7-s + (−0.847 + 0.489i)11-s + (1.20 − 0.698i)13-s + (−0.192 − 0.111i)17-s + (−0.739 − 1.28i)19-s + (−1.10 − 0.640i)23-s + (0.0798 + 0.138i)25-s + (−0.228 + 0.394i)29-s − 0.578·31-s + (−1.04 − 0.246i)35-s + (−0.671 − 1.16i)37-s + (−1.62 + 0.936i)41-s + (1.55 + 0.899i)43-s + 0.706·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.462 + 0.886i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.462 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7136378350\)
\(L(\frac12)\) \(\approx\) \(0.7136378350\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.53 - 0.763i)T \)
good5 \( 1 + (-2.08 - 1.20i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.81 - 1.62i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.35 + 2.51i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.795 + 0.459i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.22 + 5.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.31 + 3.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 - 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.21T + 31T^{2} \)
37 \( 1 + (4.08 + 7.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.3 - 5.99i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.2 - 5.89i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.84T + 47T^{2} \)
53 \( 1 + (-1.56 + 2.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.86T + 59T^{2} \)
61 \( 1 + 9.49iT - 61T^{2} \)
67 \( 1 + 0.359iT - 67T^{2} \)
71 \( 1 + 2.32iT - 71T^{2} \)
73 \( 1 + (4.01 + 2.31i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 4.36iT - 79T^{2} \)
83 \( 1 + (-8.68 + 15.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.68 + 5.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.79 + 5.65i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.614511194353622689357531432900, −7.68911859935093007926344777705, −6.77966052976195571438239374963, −6.18754524469685788934258120499, −5.65418230749416675833772011905, −4.64304464804564943927593762473, −3.51584486460922908900209796215, −2.68596413969030747469640615182, −1.97069313557448415785698011797, −0.20717732777249445587297228023, 1.38632076658355597144409114250, 2.26386839763927442300917229598, 3.55920298519177901839303947064, 4.07457217926585784914235419615, 5.43772303998995424040385211597, 5.89456721617447431191494359533, 6.48049793854220851904947389183, 7.48419352412763778774165121233, 8.394535954832268198314287664312, 8.959066591028472090389125991348

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