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Properties

Label 2-3024-252.11-c1-0-45
Degree $2$
Conductor $3024$
Sign $-0.174 + 0.984i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.05 + 1.18i)5-s + (1.79 − 1.94i)7-s + (−2.07 − 3.58i)11-s + (−1.38 − 2.40i)13-s + (−1.58 − 0.914i)17-s + (2.73 − 1.57i)19-s + (−0.510 + 0.884i)23-s + (0.306 + 0.530i)25-s + (−1.59 − 0.923i)29-s − 9.31i·31-s + (5.98 − 1.85i)35-s + (3.15 + 5.45i)37-s + (−8.53 + 4.93i)41-s + (−9.54 − 5.50i)43-s − 3.18·47-s + ⋯
L(s)  = 1  + (0.917 + 0.529i)5-s + (0.679 − 0.733i)7-s + (−0.624 − 1.08i)11-s + (−0.384 − 0.666i)13-s + (−0.384 − 0.221i)17-s + (0.626 − 0.361i)19-s + (−0.106 + 0.184i)23-s + (0.0612 + 0.106i)25-s + (−0.296 − 0.171i)29-s − 1.67i·31-s + (1.01 − 0.313i)35-s + (0.517 + 0.897i)37-s + (−1.33 + 0.769i)41-s + (−1.45 − 0.840i)43-s − 0.463·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.720185684\)
\(L(\frac12)\) \(\approx\) \(1.720185684\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 + 1.94i)T \)
good5 \( 1 + (-2.05 - 1.18i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.07 + 3.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.38 + 2.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.58 + 0.914i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.73 + 1.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.510 - 0.884i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.59 + 0.923i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.31iT - 31T^{2} \)
37 \( 1 + (-3.15 - 5.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.53 - 4.93i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.54 + 5.50i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.18T + 47T^{2} \)
53 \( 1 + (10.2 + 5.94i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.24T + 59T^{2} \)
61 \( 1 - 0.853T + 61T^{2} \)
67 \( 1 - 0.299iT - 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 + (0.419 - 0.726i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + (-2.64 + 4.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.62 - 0.940i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.03 - 13.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.174998252852058223451020428590, −8.007162685577621940502551083452, −6.93659359470529034084540504761, −6.26846167484340792704855254582, −5.39568986373421139279286919064, −4.83786327814897478230052750946, −3.59875306845961790614739479522, −2.79277606288319279176787092740, −1.83119636430184072159732708795, −0.49232254649419813516642671762, 1.66667607385573627808200365745, 1.99736984539225774257693516975, 3.21444757875337632376052900137, 4.64983561415228414956595157958, 4.99460332011990719171911716417, 5.73965973495083765490082736824, 6.65286991150640700740629864638, 7.45873694769947961831338631773, 8.281460473411511776145573533365, 8.993243612077371710771180185611

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