Properties

Degree $2$
Conductor $3024$
Sign $-0.386 - 0.922i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.92 + 3.32i)5-s + (2.55 + 0.693i)7-s + (−0.903 + 1.56i)11-s + (−0.692 + 1.19i)13-s + (0.833 + 1.44i)17-s + (0.0802 − 0.138i)19-s + (−1.60 − 2.77i)23-s + (−4.87 + 8.44i)25-s + (3.78 + 6.54i)29-s − 3.22·31-s + (2.59 + 9.82i)35-s + (1.58 − 2.74i)37-s + (−6.00 + 10.3i)41-s + (−3.45 − 5.98i)43-s + 11.4·47-s + ⋯
L(s)  = 1  + (0.858 + 1.48i)5-s + (0.965 + 0.261i)7-s + (−0.272 + 0.471i)11-s + (−0.192 + 0.332i)13-s + (0.202 + 0.350i)17-s + (0.0184 − 0.0318i)19-s + (−0.333 − 0.577i)23-s + (−0.975 + 1.68i)25-s + (0.701 + 1.21i)29-s − 0.578·31-s + (0.439 + 1.66i)35-s + (0.260 − 0.451i)37-s + (−0.937 + 1.62i)41-s + (−0.526 − 0.912i)43-s + 1.66·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.386 - 0.922i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.226512188\)
\(L(\frac12)\) \(\approx\) \(2.226512188\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 - 0.693i)T \)
good5 \( 1 + (-1.92 - 3.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.903 - 1.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.692 - 1.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.833 - 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0802 + 0.138i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.60 + 2.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.78 - 6.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.22T + 31T^{2} \)
37 \( 1 + (-1.58 + 2.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.00 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.45 + 5.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + (1.37 + 2.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 15.0T + 59T^{2} \)
61 \( 1 + 9.20T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 6.93T + 71T^{2} \)
73 \( 1 + (6.22 + 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 + (1.45 + 2.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.04 - 8.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.18 - 7.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.945010143834233598093807106990, −8.190604777962639499064596671549, −7.25572705650149307915082549465, −6.82214262234629994428786630148, −5.92643394581920588942276481674, −5.24630393837176712137918176452, −4.30567612608975001316557370422, −3.16723490275137604975258479891, −2.34124668912049794564132669395, −1.62815264106558070896990678800, 0.68625410106270524358785150689, 1.58484315051892057223864877176, 2.54643892949723033796755149978, 3.96681041282961271574511092734, 4.72032870853404723792517997413, 5.47261694272528910534137063033, 5.81875021544196545719147624453, 7.09898674373438442690001547150, 7.966485478736921604171171759847, 8.487654156024115750049990806390

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