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Properties

Label 2-3024-336.53-c0-0-3
Degree $2$
Conductor $3024$
Sign $-0.657 + 0.753i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (−0.258 − 0.965i)11-s + (−1 + i)13-s + (−0.965 + 0.258i)14-s + (0.500 + 0.866i)16-s + (−1.36 − 0.366i)19-s + i·22-s + (−0.707 − 1.22i)23-s + (−0.866 − 0.5i)25-s + (1.22 − 0.707i)26-s + 28-s + (−0.707 − 0.707i)29-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (−0.258 − 0.965i)11-s + (−1 + i)13-s + (−0.965 + 0.258i)14-s + (0.500 + 0.866i)16-s + (−1.36 − 0.366i)19-s + i·22-s + (−0.707 − 1.22i)23-s + (−0.866 − 0.5i)25-s + (1.22 − 0.707i)26-s + 28-s + (−0.707 − 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.657 + 0.753i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ -0.657 + 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5040466158\)
\(L(\frac12)\) \(\approx\) \(0.5040466158\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.581613039467562329099639121642, −8.009495712852487821556980355469, −7.38236756097404828319770879112, −6.50976316471980796764610321760, −5.82920677077697369907656607382, −4.47739056072834068925645726941, −3.96703588176040379938055162294, −2.48900880462101333229988881673, −1.97558138506320109132920793586, −0.39140652143569007224823501915, 1.67682509727456394218402982047, 2.23320531826477562168352594061, 3.46167438337546513749952591241, 4.92966034015020175278673285929, 5.29673500669524120284276370376, 6.30495011740410918737160568909, 7.14793613232703863376227694125, 7.925502430356426879537732398426, 8.196602034472543332718558180172, 9.150711174870492199809719862312

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