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Properties

Label 2-3024-63.4-c1-0-35
Degree $2$
Conductor $3024$
Sign $0.794 + 0.607i$
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  + 1.58·5-s + (1.80 − 1.93i)7-s + 5.17·11-s + (−0.681 + 1.18i)13-s + (2.30 − 3.99i)17-s + (−0.0321 − 0.0557i)19-s + 6.74·23-s − 2.49·25-s + (−4.70 − 8.15i)29-s + (−1.33 − 2.30i)31-s + (2.86 − 3.05i)35-s + (0.880 + 1.52i)37-s + (0.858 − 1.48i)41-s + (5.12 + 8.86i)43-s + (−2.60 + 4.51i)47-s + ⋯
L(s)  = 1  + 0.707·5-s + (0.683 − 0.729i)7-s + 1.55·11-s + (−0.189 + 0.327i)13-s + (0.559 − 0.969i)17-s + (−0.00738 − 0.0127i)19-s + 1.40·23-s − 0.499·25-s + (−0.874 − 1.51i)29-s + (−0.239 − 0.414i)31-s + (0.483 − 0.516i)35-s + (0.144 + 0.250i)37-s + (0.134 − 0.232i)41-s + (0.780 + 1.35i)43-s + (−0.379 + 0.657i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.632867532\)
\(L(\frac12)\) \(\approx\) \(2.632867532\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 1.93i)T \)
good5 \( 1 - 1.58T + 5T^{2} \)
11 \( 1 - 5.17T + 11T^{2} \)
13 \( 1 + (0.681 - 1.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.30 + 3.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0321 + 0.0557i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.74T + 23T^{2} \)
29 \( 1 + (4.70 + 8.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.33 + 2.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.858 + 1.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.12 - 8.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.60 - 4.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.479 + 0.831i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.66 - 8.08i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.19 - 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.24 + 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + (0.941 - 1.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.26 + 5.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.08 + 8.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.12 - 7.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.26 + 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.846117958492853530240788601561, −7.67370627383732243665424734348, −7.26742335010648946485821735926, −6.33617988929331468960107284716, −5.68222174368690564668128742064, −4.62496103427935233841868377457, −4.08987778032991241787439452160, −2.95399925244838482683138147136, −1.78194094722549470402193452124, −0.946365782672896600178612914159, 1.31141764168966272803955484920, 1.95444687710554939861871887616, 3.19860806882596439456816359526, 4.04584332298345808672355732608, 5.18736872073735150175693109342, 5.63428378349700900043019988594, 6.50152116220563159271351565746, 7.22295058252846456836800553911, 8.178383820736902133246053581553, 9.008858341231727974916655296529

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