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Properties

Label 2-3024-63.41-c1-0-10
Degree $2$
Conductor $3024$
Sign $-0.266 - 0.963i$
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  + (−1.21 + 2.10i)5-s + (−1.05 − 2.42i)7-s + (2.09 − 1.21i)11-s + (−4.73 − 2.73i)13-s − 2.58·17-s − 0.402i·19-s + (3.06 + 1.77i)23-s + (−0.440 − 0.762i)25-s + (6.31 − 3.64i)29-s + (3.63 + 2.10i)31-s + (6.37 + 0.722i)35-s − 3.19·37-s + (−4.03 + 6.99i)41-s + (4.22 + 7.31i)43-s + (−2.25 − 3.91i)47-s + ⋯
L(s)  = 1  + (−0.542 + 0.939i)5-s + (−0.399 − 0.916i)7-s + (0.632 − 0.365i)11-s + (−1.31 − 0.758i)13-s − 0.625·17-s − 0.0923i·19-s + (0.639 + 0.369i)23-s + (−0.0880 − 0.152i)25-s + (1.17 − 0.677i)29-s + (0.653 + 0.377i)31-s + (1.07 + 0.122i)35-s − 0.525·37-s + (−0.630 + 1.09i)41-s + (0.644 + 1.11i)43-s + (−0.329 − 0.570i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8062588862\)
\(L(\frac12)\) \(\approx\) \(0.8062588862\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.05 + 2.42i)T \)
good5 \( 1 + (1.21 - 2.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.09 + 1.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.73 + 2.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.58T + 17T^{2} \)
19 \( 1 + 0.402iT - 19T^{2} \)
23 \( 1 + (-3.06 - 1.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.31 + 3.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.63 - 2.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 + (4.03 - 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.22 - 7.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.25 + 3.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 14.0iT - 53T^{2} \)
59 \( 1 + (0.0779 - 0.134i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.2 - 5.90i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.53 - 4.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.73iT - 71T^{2} \)
73 \( 1 - 8.80iT - 73T^{2} \)
79 \( 1 + (5.66 + 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.50 - 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (-4.97 + 2.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.990079889826869196314272720215, −7.948294324777618516102948569538, −7.41956222473696784166943116367, −6.74770951446926225391893105612, −6.18976158938120852065742301356, −4.90738932513651922495205509750, −4.23523193402814784037065631293, −3.19120995704559895761235544250, −2.74809405640372834828682643024, −1.05467922869144439050684399058, 0.28791609949281964360567405605, 1.79405597033120230916445193132, 2.70564124816671449958704657629, 3.86938753745821263617895274678, 4.79045425595066457637671557092, 5.10829478302845168112574669742, 6.41491046456386771520359892852, 6.87622034136797613167720929400, 7.84175293527124081225751810090, 8.797017344314822326668734220143

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