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Properties

Label 2-3024-63.4-c1-0-44
Degree $2$
Conductor $3024$
Sign $-0.282 + 0.959i$
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  + 1.90·5-s + (2.43 − 1.03i)7-s − 3.06·11-s + (1.13 − 1.96i)13-s + (0.713 − 1.23i)17-s + (−2.98 − 5.16i)19-s − 7.15·23-s − 1.37·25-s + (−0.468 − 0.810i)29-s + (−4.11 − 7.11i)31-s + (4.63 − 1.97i)35-s + (−1.41 − 2.45i)37-s + (5.31 − 9.20i)41-s + (−2.98 − 5.16i)43-s + (0.483 − 0.837i)47-s + ⋯
L(s)  = 1  + 0.851·5-s + (0.920 − 0.391i)7-s − 0.924·11-s + (0.313 − 0.543i)13-s + (0.173 − 0.299i)17-s + (−0.684 − 1.18i)19-s − 1.49·23-s − 0.275·25-s + (−0.0869 − 0.150i)29-s + (−0.738 − 1.27i)31-s + (0.782 − 0.333i)35-s + (−0.232 − 0.403i)37-s + (0.830 − 1.43i)41-s + (−0.455 − 0.788i)43-s + (0.0705 − 0.122i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.282 + 0.959i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.282 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.642124664\)
\(L(\frac12)\) \(\approx\) \(1.642124664\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.43 + 1.03i)T \)
good5 \( 1 - 1.90T + 5T^{2} \)
11 \( 1 + 3.06T + 11T^{2} \)
13 \( 1 + (-1.13 + 1.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.713 + 1.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.98 + 5.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.15T + 23T^{2} \)
29 \( 1 + (0.468 + 0.810i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.11 + 7.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.41 + 2.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.31 + 9.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.98 + 5.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.483 + 0.837i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.45 - 9.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.68 - 9.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.449 - 0.778i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.813 - 1.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.36T + 71T^{2} \)
73 \( 1 + (0.996 - 1.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.16 - 7.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.98 + 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.58 - 4.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.922 - 1.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.505568046824322435724346335559, −7.68044178617209322943543965180, −7.20754503885451419265259744048, −5.90506397288971646133413079659, −5.63018268677309014460327758126, −4.64319638082272378099343752709, −3.85636909302598041982995014456, −2.50145597122954899804131503559, −1.93058172251229350548771134690, −0.46304735046014043392314662674, 1.65509976602178071126935339589, 2.04528050264768096572331495373, 3.32163977642385015716527933183, 4.38073776795982465304269523787, 5.18824392341982852051517661891, 5.91275397042632709000468535380, 6.44774863343495796857634289818, 7.68846311041645697533717053264, 8.182424061971468352643948638470, 8.811087994252212388329350345249

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