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Properties

Label 2-3024-36.11-c1-0-35
Degree $2$
Conductor $3024$
Sign $-0.758 + 0.651i$
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  + (1.84 − 1.06i)5-s + (0.866 + 0.5i)7-s + (−0.546 + 0.945i)11-s + (−3.09 − 5.35i)13-s − 3.15i·17-s + 2.17i·19-s + (−4.09 − 7.09i)23-s + (−0.231 + 0.400i)25-s + (−5.07 − 2.92i)29-s + (−4.70 + 2.71i)31-s + 2.13·35-s − 8.66·37-s + (−3.56 + 2.05i)41-s + (5.70 + 3.29i)43-s + (−0.871 + 1.51i)47-s + ⋯
L(s)  = 1  + (0.825 − 0.476i)5-s + (0.327 + 0.188i)7-s + (−0.164 + 0.285i)11-s + (−0.857 − 1.48i)13-s − 0.764i·17-s + 0.498i·19-s + (−0.854 − 1.48i)23-s + (−0.0462 + 0.0800i)25-s + (−0.941 − 0.543i)29-s + (−0.845 + 0.487i)31-s + 0.360·35-s − 1.42·37-s + (−0.556 + 0.321i)41-s + (0.869 + 0.502i)43-s + (−0.127 + 0.220i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.758 + 0.651i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.098571095\)
\(L(\frac12)\) \(\approx\) \(1.098571095\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-1.84 + 1.06i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.546 - 0.945i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.09 + 5.35i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.15iT - 17T^{2} \)
19 \( 1 - 2.17iT - 19T^{2} \)
23 \( 1 + (4.09 + 7.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.07 + 2.92i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.70 - 2.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.66T + 37T^{2} \)
41 \( 1 + (3.56 - 2.05i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.70 - 3.29i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.871 - 1.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.10iT - 53T^{2} \)
59 \( 1 + (6.22 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.42 - 4.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.27 + 4.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.62T + 71T^{2} \)
73 \( 1 + 1.05T + 73T^{2} \)
79 \( 1 + (-11.2 - 6.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.54 - 2.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.37iT - 89T^{2} \)
97 \( 1 + (-3.23 + 5.61i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.328180052752841624318306158143, −7.82983437784410348470158977516, −6.95801694356658297279303020059, −5.97882368198954400250753564794, −5.27882196154921495697159691443, −4.84805447063870094869789557148, −3.60299564461644368355884611878, −2.52846841150373685586017810467, −1.76256375008092555505981537460, −0.30461661978482353032644394638, 1.71137045638395843256516084653, 2.19434810899369809467492647107, 3.50155169065471455733139475693, 4.27950500865231108625347429439, 5.34242433257282787688804700866, 5.90062339819301448006157840956, 6.87179177290361197088494274938, 7.33566463624384424344806544121, 8.257427861453970420644068849903, 9.317472989703653333714125535895

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