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Properties

Label 2-3024-63.47-c1-0-11
Degree $2$
Conductor $3024$
Sign $0.583 - 0.812i$
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  − 2.20·5-s + (−2.16 + 1.52i)7-s + 0.181i·11-s + (2.50 − 1.44i)13-s + (−1.98 − 3.43i)17-s + (−0.867 − 0.500i)19-s − 5.61i·23-s − 0.121·25-s + (−0.703 − 0.406i)29-s + (6.89 + 3.98i)31-s + (4.78 − 3.35i)35-s + (1.25 − 2.17i)37-s + (0.612 + 1.06i)41-s + (−5.47 + 9.48i)43-s + (3.57 + 6.19i)47-s + ⋯
L(s)  = 1  − 0.987·5-s + (−0.818 + 0.574i)7-s + 0.0548i·11-s + (0.694 − 0.400i)13-s + (−0.481 − 0.833i)17-s + (−0.198 − 0.114i)19-s − 1.17i·23-s − 0.0242·25-s + (−0.130 − 0.0754i)29-s + (1.23 + 0.715i)31-s + (0.808 − 0.567i)35-s + (0.206 − 0.357i)37-s + (0.0957 + 0.165i)41-s + (−0.835 + 1.44i)43-s + (0.521 + 0.903i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.583 - 0.812i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.583 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9764473337\)
\(L(\frac12)\) \(\approx\) \(0.9764473337\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 - 1.52i)T \)
good5 \( 1 + 2.20T + 5T^{2} \)
11 \( 1 - 0.181iT - 11T^{2} \)
13 \( 1 + (-2.50 + 1.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.98 + 3.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.867 + 0.500i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.61iT - 23T^{2} \)
29 \( 1 + (0.703 + 0.406i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.89 - 3.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.25 + 2.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.612 - 1.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.47 - 9.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.57 - 6.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.75 + 1.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.27 - 5.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.97 + 4.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.44 + 5.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (10.1 - 5.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.35 - 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.19 - 12.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.11 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.01 + 1.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.619892179700953079655311036506, −8.285390767199443929443819430488, −7.30274733538263336421056280375, −6.57640410226512936531227399631, −5.93352817150214406236889095246, −4.85448887126264137304240865764, −4.12095777331257211208944330552, −3.17701743176154356361826739731, −2.49053564770919420298501951589, −0.795927124752326325546827456651, 0.44118344381431285046149198956, 1.83602139187722444549175275040, 3.24216229295953712847515865498, 3.84886790987139414587630427530, 4.42539105162523471196013380905, 5.68572282839974458614773027619, 6.42984563927230116496332041806, 7.12161941945181604642135253946, 7.86152785428672049014920741112, 8.541568013561318782421817429033

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