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Properties

Label 2-3024-63.20-c1-0-35
Degree $2$
Conductor $3024$
Sign $0.672 + 0.739i$
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  + (−0.0977 − 0.169i)5-s + (2.48 − 0.900i)7-s + (1.54 + 0.890i)11-s + (5.11 − 2.95i)13-s + 0.588·17-s − 2.48i·19-s + (3.85 − 2.22i)23-s + (2.48 − 4.29i)25-s + (−6.28 − 3.62i)29-s + (−3.61 + 2.08i)31-s + (−0.395 − 0.333i)35-s − 2.57·37-s + (0.311 + 0.540i)41-s + (−5.08 + 8.80i)43-s + (−3.57 + 6.19i)47-s + ⋯
L(s)  = 1  + (−0.0437 − 0.0757i)5-s + (0.940 − 0.340i)7-s + (0.464 + 0.268i)11-s + (1.41 − 0.819i)13-s + 0.142·17-s − 0.570i·19-s + (0.804 − 0.464i)23-s + (0.496 − 0.859i)25-s + (−1.16 − 0.673i)29-s + (−0.649 + 0.374i)31-s + (−0.0668 − 0.0563i)35-s − 0.422·37-s + (0.0487 + 0.0843i)41-s + (−0.775 + 1.34i)43-s + (−0.521 + 0.903i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.672 + 0.739i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.672 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.269440785\)
\(L(\frac12)\) \(\approx\) \(2.269440785\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.48 + 0.900i)T \)
good5 \( 1 + (0.0977 + 0.169i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.54 - 0.890i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.11 + 2.95i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.588T + 17T^{2} \)
19 \( 1 + 2.48iT - 19T^{2} \)
23 \( 1 + (-3.85 + 2.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.28 + 3.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.61 - 2.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.57T + 37T^{2} \)
41 \( 1 + (-0.311 - 0.540i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.08 - 8.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.57 - 6.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + (-5.75 - 9.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.57 - 3.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.927 + 1.60i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.58iT - 71T^{2} \)
73 \( 1 + 5.66iT - 73T^{2} \)
79 \( 1 + (2.92 - 5.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.740 - 1.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.44T + 89T^{2} \)
97 \( 1 + (-0.0722 - 0.0417i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.513706013184255823430596237379, −8.017497832601692898248639107397, −7.13719998342165190234921603862, −6.39455685351454339608036169609, −5.50444029340302407941889284225, −4.72799755592790763062628198968, −3.93737491007684407507064712202, −3.03145701937758145809393562690, −1.75892183845232795413612528405, −0.815217302472283630426351411156, 1.27314686689866948356544138531, 1.95735462394464739689971444713, 3.45819916751489454971453536005, 3.91887233456268243910041268713, 5.14465623037521316450408528294, 5.63163854774617643533654744839, 6.62538507603039195624231345373, 7.29374924061062155506651777863, 8.181528355931212288937269983036, 8.915631371021099158475670812483

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