Properties

Degree $2$
Conductor $3024$
Sign $-0.477 + 0.878i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.891 − 1.54i)5-s + (2.54 − 0.727i)7-s + (2.80 − 4.86i)11-s + (3.14 − 5.43i)13-s + (−0.646 − 1.11i)17-s + (−0.559 + 0.968i)19-s + (−3.80 − 6.59i)23-s + (0.909 − 1.57i)25-s + (1.57 + 2.72i)29-s − 1.00·31-s + (−3.39 − 3.28i)35-s + (−5.96 + 10.3i)37-s + (−4.14 + 7.17i)41-s + (−2.34 − 4.06i)43-s − 1.94·47-s + ⋯
L(s)  = 1  + (−0.398 − 0.690i)5-s + (0.961 − 0.274i)7-s + (0.846 − 1.46i)11-s + (0.870 − 1.50i)13-s + (−0.156 − 0.271i)17-s + (−0.128 + 0.222i)19-s + (−0.794 − 1.37i)23-s + (0.181 − 0.315i)25-s + (0.292 + 0.506i)29-s − 0.180·31-s + (−0.573 − 0.554i)35-s + (−0.980 + 1.69i)37-s + (−0.646 + 1.12i)41-s + (−0.358 − 0.620i)43-s − 0.283·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.477 + 0.878i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.878005816\)
\(L(\frac12)\) \(\approx\) \(1.878005816\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 + 0.727i)T \)
good5 \( 1 + (0.891 + 1.54i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.80 + 4.86i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.14 + 5.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.646 + 1.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.559 - 0.968i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.80 + 6.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.57 - 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.00T + 31T^{2} \)
37 \( 1 + (5.96 - 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.14 - 7.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.34 + 4.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.94T + 47T^{2} \)
53 \( 1 + (-4.45 - 7.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.38T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 - 2.55T + 67T^{2} \)
71 \( 1 + 8.86T + 71T^{2} \)
73 \( 1 + (-5.67 - 9.83i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + (1.60 + 2.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.404 + 0.700i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.10 - 1.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.342287673336571741847075758979, −8.234314481125076699615828868141, −6.94872507879497623038928855117, −6.12424195496193307051917989602, −5.38128792823839302559794448181, −4.56338622449235287709749521399, −3.76376973799933623843848308496, −2.91595421218207479299642531281, −1.36687567938574967645167213477, −0.63235063279068731352036499435, 1.66272526156550039279208056807, 2.06965122266451863712809211016, 3.71201261776141301185649787476, 4.09275640524191393304014198913, 5.04190315415987498120157222124, 6.00713118286438984885307356860, 7.06503001458374396706603973970, 7.13804494269384581215918843503, 8.253267926155534021065847085011, 8.974024437815440667264068682739

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