Properties

Label 2-3024-63.20-c1-0-34
Degree $2$
Conductor $3024$
Sign $0.0755 + 0.997i$
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  + (1.11 + 1.92i)5-s + (−1.78 − 1.95i)7-s + (−1.51 − 0.876i)11-s + (3.37 − 1.94i)13-s − 1.78·17-s − 2.73i·19-s + (1.64 − 0.947i)23-s + (0.0196 − 0.0339i)25-s + (7.48 + 4.32i)29-s + (−7.18 + 4.14i)31-s + (1.78 − 5.61i)35-s − 9.91·37-s + (−3.42 − 5.93i)41-s + (4.22 − 7.32i)43-s + (−1.47 + 2.54i)47-s + ⋯
L(s)  = 1  + (0.498 + 0.862i)5-s + (−0.673 − 0.738i)7-s + (−0.457 − 0.264i)11-s + (0.935 − 0.540i)13-s − 0.433·17-s − 0.628i·19-s + (0.342 − 0.197i)23-s + (0.00392 − 0.00678i)25-s + (1.39 + 0.802i)29-s + (−1.29 + 0.745i)31-s + (0.301 − 0.949i)35-s − 1.63·37-s + (−0.535 − 0.927i)41-s + (0.644 − 1.11i)43-s + (−0.214 + 0.371i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0755 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0755 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.0755 + 0.997i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.0755 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.310420747\)
\(L(\frac12)\) \(\approx\) \(1.310420747\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.78 + 1.95i)T \)
good5 \( 1 + (-1.11 - 1.92i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.51 + 0.876i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.37 + 1.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.78T + 17T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 + (-1.64 + 0.947i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.48 - 4.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.18 - 4.14i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.91T + 37T^{2} \)
41 \( 1 + (3.42 + 5.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.22 + 7.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.47 - 2.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.45iT - 53T^{2} \)
59 \( 1 + (-0.449 - 0.778i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.3 + 5.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.38 + 9.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 - 4.71iT - 73T^{2} \)
79 \( 1 + (-4.70 + 8.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.326 + 0.565i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.70T + 89T^{2} \)
97 \( 1 + (-0.0294 - 0.0169i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.703017396238953149662227891139, −7.65814074004242203079364871064, −6.83088278084310211605339432926, −6.51760084921996640901990257852, −5.56720973706895160350066636690, −4.70105869306826523046511529150, −3.41439690486347638428455145040, −3.12922315998917384013932670525, −1.86750488733206491343870056679, −0.41657389995443660920423668082, 1.26444175072654481982041281223, 2.23517525132952613864515571686, 3.27393147878357634071753208148, 4.28551464895448491423534057561, 5.14340889405746739175353428616, 5.87879281755327244368221841134, 6.43999974245807205564564124404, 7.41038994307098504946910819052, 8.437384816028051458568729721068, 8.861095783076613755459976326101

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