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Properties

Label 2-3024-63.59-c1-0-35
Degree $2$
Conductor $3024$
Sign $0.985 + 0.170i$
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  + 2.22·5-s + (2.45 + 0.996i)7-s + 1.17i·11-s + (3.12 + 1.80i)13-s + (3.71 − 6.42i)17-s + (3.05 − 1.76i)19-s − 5.81i·23-s − 0.0674·25-s + (6.04 − 3.48i)29-s + (−6.88 + 3.97i)31-s + (5.44 + 2.21i)35-s + (−5.54 − 9.60i)37-s + (0.809 − 1.40i)41-s + (−0.904 − 1.56i)43-s + (−4.26 + 7.38i)47-s + ⋯
L(s)  = 1  + 0.993·5-s + (0.926 + 0.376i)7-s + 0.353i·11-s + (0.866 + 0.500i)13-s + (0.900 − 1.55i)17-s + (0.700 − 0.404i)19-s − 1.21i·23-s − 0.0134·25-s + (1.12 − 0.648i)29-s + (−1.23 + 0.713i)31-s + (0.920 + 0.374i)35-s + (−0.911 − 1.57i)37-s + (0.126 − 0.218i)41-s + (−0.137 − 0.238i)43-s + (−0.622 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.985 + 0.170i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.985 + 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.795531716\)
\(L(\frac12)\) \(\approx\) \(2.795531716\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 0.996i)T \)
good5 \( 1 - 2.22T + 5T^{2} \)
11 \( 1 - 1.17iT - 11T^{2} \)
13 \( 1 + (-3.12 - 1.80i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.71 + 6.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.05 + 1.76i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.81iT - 23T^{2} \)
29 \( 1 + (-6.04 + 3.48i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.88 - 3.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.54 + 9.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.809 + 1.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.904 + 1.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.26 - 7.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.62 - 5.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.00 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.09 + 4.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.96 - 8.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.67iT - 71T^{2} \)
73 \( 1 + (-6.92 - 3.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.25 - 3.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.390 + 0.677i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.75 + 3.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.49 - 2.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.945521519393453872291504560800, −7.942282588826034229890414647459, −7.20211894821487383172376280347, −6.40603371830995532459555005491, −5.47461653245459349431016539202, −5.07664472341080791770944605546, −4.07325244272853256335059952177, −2.83402100084977787948909666431, −2.04099387791307428993135154430, −1.03056095494188417604792433165, 1.28919902023570974303326764436, 1.74281265770877790555844388291, 3.25144076046984088851752435141, 3.86639093707775174442392987746, 5.16528924860453204687534592678, 5.62442558581537249218177009042, 6.30001311725548623109680012382, 7.32288522950317289378970349226, 8.167933032676389855419161348221, 8.522989075305884110544382628645

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