Properties

Degree $2$
Conductor $3024$
Sign $0.738 - 0.674i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0309 − 0.0536i)5-s + (0.981 + 2.45i)7-s + (1.59 − 2.75i)11-s + (−0.252 + 0.437i)13-s + (0.554 + 0.960i)17-s + (−0.933 + 1.61i)19-s + (3.10 + 5.37i)23-s + (2.49 − 4.32i)25-s + (−2.39 − 4.15i)29-s + 2.53·31-s + (0.101 − 0.128i)35-s + (−4.26 + 7.38i)37-s + (4.94 − 8.56i)41-s + (3.95 + 6.85i)43-s + 6.58·47-s + ⋯
L(s)  = 1  + (−0.0138 − 0.0240i)5-s + (0.371 + 0.928i)7-s + (0.479 − 0.830i)11-s + (−0.0700 + 0.121i)13-s + (0.134 + 0.233i)17-s + (−0.214 + 0.370i)19-s + (0.646 + 1.12i)23-s + (0.499 − 0.865i)25-s + (−0.445 − 0.770i)29-s + 0.455·31-s + (0.0171 − 0.0217i)35-s + (−0.700 + 1.21i)37-s + (0.772 − 1.33i)41-s + (0.603 + 1.04i)43-s + 0.960·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.738 - 0.674i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.738 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.949344340\)
\(L(\frac12)\) \(\approx\) \(1.949344340\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.981 - 2.45i)T \)
good5 \( 1 + (0.0309 + 0.0536i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.59 + 2.75i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.252 - 0.437i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.554 - 0.960i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.933 - 1.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.10 - 5.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.39 + 4.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.53T + 31T^{2} \)
37 \( 1 + (4.26 - 7.38i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.94 + 8.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.95 - 6.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.58T + 47T^{2} \)
53 \( 1 + (-1.58 - 2.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.01T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 3.33T + 67T^{2} \)
71 \( 1 - 2.25T + 71T^{2} \)
73 \( 1 + (-2.07 - 3.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 2.97T + 79T^{2} \)
83 \( 1 + (-2.17 - 3.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.30 + 7.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.27 + 5.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.851996125953842252871388829302, −8.147968782880034593313115136346, −7.42065644221988520959133585947, −6.34129220179858774446102428013, −5.85140814781197938283019003755, −5.03107222577758386661111603312, −4.09313481975711395448661430979, −3.15231562349684019750063945475, −2.21963999400323508063114466859, −1.06411704404200740913243688375, 0.74003582493394345573528459454, 1.85478120030939707437208809223, 3.00008964498267561224187929831, 4.03356443758291003583925733175, 4.66130873464405901858500257307, 5.45124231256997153557281193074, 6.61116512880476685368293474497, 7.15140686685614881794384620544, 7.69933692306248319381927424296, 8.753772802091611086992920226577

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