Properties

Degree $2$
Conductor $3024$
Sign $0.0907 - 0.995i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0731 − 0.126i)5-s + (2.33 + 1.25i)7-s + (−0.832 + 1.44i)11-s + (0.0999 − 0.173i)13-s + (−3.13 − 5.43i)17-s + (−3.45 + 5.99i)19-s + (3.09 + 5.35i)23-s + (2.48 − 4.31i)25-s + (2.46 + 4.27i)29-s + 2.51·31-s + (−0.0117 − 0.386i)35-s + (−3.50 + 6.06i)37-s + (−1.15 + 2.00i)41-s + (0.940 + 1.62i)43-s − 1.81·47-s + ⋯
L(s)  = 1  + (−0.0327 − 0.0566i)5-s + (0.880 + 0.473i)7-s + (−0.250 + 0.434i)11-s + (0.0277 − 0.0480i)13-s + (−0.760 − 1.31i)17-s + (−0.793 + 1.37i)19-s + (0.644 + 1.11i)23-s + (0.497 − 0.862i)25-s + (0.458 + 0.793i)29-s + 0.452·31-s + (−0.00198 − 0.0653i)35-s + (−0.575 + 0.996i)37-s + (−0.180 + 0.313i)41-s + (0.143 + 0.248i)43-s − 0.264·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0907 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0907 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.0907 - 0.995i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.0907 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.599696587\)
\(L(\frac12)\) \(\approx\) \(1.599696587\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.33 - 1.25i)T \)
good5 \( 1 + (0.0731 + 0.126i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.832 - 1.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0999 + 0.173i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.13 + 5.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.45 - 5.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.09 - 5.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.46 - 4.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.51T + 31T^{2} \)
37 \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.15 - 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.940 - 1.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.81T + 47T^{2} \)
53 \( 1 + (-2.67 - 4.62i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.57T + 59T^{2} \)
61 \( 1 + 0.678T + 61T^{2} \)
67 \( 1 - 6.18T + 67T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 + (0.778 + 1.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + (-3.75 - 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.53 - 7.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.98 + 6.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.736244934746480829909372994538, −8.265355112020781637319732506389, −7.43059815319711104037402524087, −6.70859767837390894633499685310, −5.77980285067461623495364764386, −4.92209332145405146868915885996, −4.46772964157523049547235688484, −3.19927393944202618957259521874, −2.28213849382436919798826369214, −1.28233338073365597384685054339, 0.51704283684771202610820425109, 1.83448911239918022148338744024, 2.76128202915823273488896191643, 3.97110655606360671875348524079, 4.58967570615710924991018324094, 5.37868433851488537335833218532, 6.43678124143901692103605879793, 6.96087552020993413451770500010, 7.891089997509977553262235256480, 8.637841327977670890917458781170

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