Properties

Degree $2$
Conductor $3024$
Sign $-0.999 - 0.0360i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.05 − 1.82i)5-s + (−2.58 − 0.569i)7-s + (−0.199 − 0.345i)11-s + (1.44 + 2.49i)13-s + (0.176 − 0.305i)17-s + (−2.84 − 4.93i)19-s + (0.438 − 0.759i)23-s + (0.285 + 0.494i)25-s + (−0.874 + 1.51i)29-s − 9.13·31-s + (−3.75 + 4.11i)35-s + (−3.39 − 5.88i)37-s + (−1.20 − 2.08i)41-s + (−0.276 + 0.479i)43-s + 11.7·47-s + ⋯
L(s)  = 1  + (0.470 − 0.815i)5-s + (−0.976 − 0.215i)7-s + (−0.0601 − 0.104i)11-s + (0.400 + 0.693i)13-s + (0.0428 − 0.0741i)17-s + (−0.653 − 1.13i)19-s + (0.0914 − 0.158i)23-s + (0.0571 + 0.0989i)25-s + (−0.162 + 0.281i)29-s − 1.64·31-s + (−0.634 + 0.694i)35-s + (−0.558 − 0.966i)37-s + (−0.187 − 0.325i)41-s + (−0.0422 + 0.0730i)43-s + 1.71·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 - 0.0360i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.999 - 0.0360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4493753326\)
\(L(\frac12)\) \(\approx\) \(0.4493753326\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 + 0.569i)T \)
good5 \( 1 + (-1.05 + 1.82i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.199 + 0.345i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.44 - 2.49i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.176 + 0.305i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.84 + 4.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.438 + 0.759i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.874 - 1.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.13T + 31T^{2} \)
37 \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.20 + 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.276 - 0.479i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + (-2.07 + 3.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.32T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 1.20T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (-0.315 + 0.546i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.48T + 79T^{2} \)
83 \( 1 + (4.59 - 7.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.29 + 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.84 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.704030531609387525970570239980, −7.41291438812431198926899776470, −6.88330103231805606872790882854, −6.00414887181834744220525113154, −5.34674281181596568139857496318, −4.40043438652968312537174485708, −3.62675813191454549434750676559, −2.54138613183984171331035041022, −1.45264333878196984424594239102, −0.13415835525979145178045549974, 1.63146755202638575723389295462, 2.74336627248483042931256919615, 3.37759854078496076540809397595, 4.28742190554829931932499875693, 5.69100459861051050842443121496, 5.93980783622465276829508653000, 6.80437651264988911855962179379, 7.48298618270880804547383517186, 8.397835302166089808076930979390, 9.160527836716706099983120752152

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