Properties

Degree $2$
Conductor $3024$
Sign $0.574 - 0.818i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.76 − 3.05i)5-s + (2.63 + 0.176i)7-s + (1.16 − 2.00i)11-s + (−2.35 + 4.08i)13-s + (0.636 + 1.10i)17-s + (−2.78 + 4.82i)19-s + (1.64 + 2.85i)23-s + (−3.72 + 6.45i)25-s + (4.32 + 7.48i)29-s − 8.51·31-s + (−4.11 − 8.38i)35-s + (−2.84 + 4.91i)37-s + (−1.66 + 2.88i)41-s + (−0.0444 − 0.0769i)43-s + 7.05·47-s + ⋯
L(s)  = 1  + (−0.789 − 1.36i)5-s + (0.997 + 0.0666i)7-s + (0.349 − 0.605i)11-s + (−0.654 + 1.13i)13-s + (0.154 + 0.267i)17-s + (−0.638 + 1.10i)19-s + (0.343 + 0.595i)23-s + (−0.745 + 1.29i)25-s + (0.802 + 1.38i)29-s − 1.52·31-s + (−0.696 − 1.41i)35-s + (−0.466 + 0.808i)37-s + (−0.260 + 0.450i)41-s + (−0.00677 − 0.0117i)43-s + 1.02·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.574 - 0.818i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.574 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.248592505\)
\(L(\frac12)\) \(\approx\) \(1.248592505\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.176i)T \)
good5 \( 1 + (1.76 + 3.05i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.16 + 2.00i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.35 - 4.08i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.636 - 1.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.78 - 4.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.64 - 2.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.32 - 7.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.51T + 31T^{2} \)
37 \( 1 + (2.84 - 4.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.66 - 2.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0444 + 0.0769i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.05T + 47T^{2} \)
53 \( 1 + (3.41 + 5.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.99T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 6.12T + 67T^{2} \)
71 \( 1 - 1.30T + 71T^{2} \)
73 \( 1 + (-6.64 - 11.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + (5.90 + 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.561 - 0.972i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.50 + 6.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.748612314658585297862402275201, −8.229612887667171240408618324593, −7.52512778311415171094921627921, −6.66472370321343122886170504150, −5.50794920794988765809061944391, −4.95093671459087245345067344794, −4.21477604852669744589703026029, −3.52087189607051571978328354730, −1.88424475779061611795279947490, −1.17268195338801899647241083324, 0.42532533593410414639050225306, 2.17228794257074146475264014334, 2.83757559735768749112155736951, 3.89238944572900527169745252983, 4.62392746425220886702704371447, 5.49191319723327847700256588627, 6.56861993821677176481970892668, 7.27776190429228578552706052140, 7.63997531800207064300330672348, 8.442212711987030068594442501939

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