Properties

Degree $2$
Conductor $3024$
Sign $0.952 + 0.303i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.296 − 0.514i)5-s + (−2.32 + 1.26i)7-s + (0.296 + 0.514i)11-s + (−1.25 − 2.17i)13-s + (−1.46 + 2.52i)17-s + (−2.69 − 4.66i)19-s + (−2.23 + 3.86i)23-s + (2.32 + 4.02i)25-s + (3.09 − 5.36i)29-s + 7.86·31-s + (−0.0394 + 1.56i)35-s + (0.5 + 0.866i)37-s + (0.136 + 0.236i)41-s + (5.58 − 9.66i)43-s + 12.1·47-s + ⋯
L(s)  = 1  + (0.132 − 0.229i)5-s + (−0.878 + 0.478i)7-s + (0.0894 + 0.154i)11-s + (−0.348 − 0.603i)13-s + (−0.354 + 0.613i)17-s + (−0.617 − 1.06i)19-s + (−0.465 + 0.805i)23-s + (0.464 + 0.804i)25-s + (0.575 − 0.996i)29-s + 1.41·31-s + (−0.00667 + 0.265i)35-s + (0.0821 + 0.142i)37-s + (0.0213 + 0.0369i)41-s + (0.851 − 1.47i)43-s + 1.77·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.952 + 0.303i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.952 + 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.471994188\)
\(L(\frac12)\) \(\approx\) \(1.471994188\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.32 - 1.26i)T \)
good5 \( 1 + (-0.296 + 0.514i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.296 - 0.514i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.25 + 2.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.46 - 2.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 - 3.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.09 + 5.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.86T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.136 - 0.236i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.58 + 9.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + (4.02 - 6.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.64T + 59T^{2} \)
61 \( 1 + 6.64T + 61T^{2} \)
67 \( 1 - 1.91T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (-3.95 + 6.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.24T + 79T^{2} \)
83 \( 1 + (-3.85 + 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.21 - 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.86 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.921163440043255968668946226647, −7.942898443065429431069670792982, −7.16295627060365298377443899355, −6.31676446893070470482699343620, −5.75994519713805954150436455801, −4.82182293506515281196490298429, −3.96796592622568655131583711415, −2.92538060991097997387531344602, −2.18169736033481201028030155186, −0.65465749046276465582053241951, 0.798908989504580595444235234905, 2.27647777965153986903325256728, 3.05615273573174483146746531650, 4.12635961907907185290580165936, 4.69237006654212370339875906609, 5.98304857728073451251955354572, 6.49186026059803636495193067715, 7.10008102522931742529605483257, 8.035681537865472219025176082281, 8.787540581852647553425799711511

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