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Properties

Label 2-3024-63.59-c1-0-44
Degree $2$
Conductor $3024$
Sign $-0.746 + 0.665i$
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  − 0.0764·5-s + (2.39 + 1.11i)7-s − 5.38i·11-s + (−4.60 − 2.65i)13-s + (1.89 − 3.27i)17-s + (4.33 − 2.50i)19-s − 2.33i·23-s − 4.99·25-s + (−8.84 + 5.10i)29-s + (−4.97 + 2.87i)31-s + (−0.183 − 0.0852i)35-s + (0.354 + 0.613i)37-s + (−3.29 + 5.71i)41-s + (−0.716 − 1.24i)43-s + (−1.46 + 2.53i)47-s + ⋯
L(s)  = 1  − 0.0341·5-s + (0.906 + 0.421i)7-s − 1.62i·11-s + (−1.27 − 0.737i)13-s + (0.458 − 0.794i)17-s + (0.995 − 0.574i)19-s − 0.487i·23-s − 0.998·25-s + (−1.64 + 0.948i)29-s + (−0.893 + 0.516i)31-s + (−0.0309 − 0.0144i)35-s + (0.0582 + 0.100i)37-s + (−0.515 + 0.892i)41-s + (−0.109 − 0.189i)43-s + (−0.213 + 0.369i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9781616878\)
\(L(\frac12)\) \(\approx\) \(0.9781616878\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.39 - 1.11i)T \)
good5 \( 1 + 0.0764T + 5T^{2} \)
11 \( 1 + 5.38iT - 11T^{2} \)
13 \( 1 + (4.60 + 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.89 + 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.33 + 2.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.33iT - 23T^{2} \)
29 \( 1 + (8.84 - 5.10i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.97 - 2.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.354 - 0.613i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.29 - 5.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.716 + 1.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.46 - 2.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.4 + 6.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.289 + 0.502i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.40 + 1.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.63 - 4.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.32iT - 71T^{2} \)
73 \( 1 + (6.17 + 3.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.469 + 0.812i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.49 + 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.51 - 2.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.18 + 3.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.379667051593832453736786029244, −7.69331431362374163501848336459, −7.16663620950770219567638189631, −5.91102097123890018491285425169, −5.33629634822286355124082040300, −4.80511848825048650173159683011, −3.42138258113242450830402486833, −2.85240164779667676317572450154, −1.61319295372920625534786404818, −0.28598368082509999770126128429, 1.68641326008248752025305753773, 2.08769608668500248445719113615, 3.68429094454654781376115767942, 4.32095888211847032426409757051, 5.11769806253094106218096166361, 5.83116147540876060547530286731, 7.06957239342320110212412572540, 7.58332299841045726721257562777, 7.893251690353578518011252314762, 9.244985981727372293255038181121

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