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Properties

Label 2-3024-252.103-c1-0-35
Degree $2$
Conductor $3024$
Sign $-0.716 + 0.697i$
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  + (−2.43 + 1.40i)5-s + (−0.717 − 2.54i)7-s + (−2.29 − 1.32i)11-s + (4.59 + 2.65i)13-s + (2.35 − 1.36i)17-s + (0.274 − 0.474i)19-s + (−1.98 + 1.14i)23-s + (1.45 − 2.51i)25-s + (3.72 + 6.45i)29-s + 2.76·31-s + (5.32 + 5.19i)35-s + (1.81 − 3.13i)37-s + (−7.12 − 4.11i)41-s + (−3.93 + 2.27i)43-s − 11.2·47-s + ⋯
L(s)  = 1  + (−1.08 + 0.628i)5-s + (−0.271 − 0.962i)7-s + (−0.690 − 0.398i)11-s + (1.27 + 0.735i)13-s + (0.571 − 0.329i)17-s + (0.0628 − 0.108i)19-s + (−0.413 + 0.238i)23-s + (0.290 − 0.503i)25-s + (0.692 + 1.19i)29-s + 0.497·31-s + (0.900 + 0.877i)35-s + (0.298 − 0.516i)37-s + (−1.11 − 0.642i)41-s + (−0.600 + 0.346i)43-s − 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.716 + 0.697i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4646555547\)
\(L(\frac12)\) \(\approx\) \(0.4646555547\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.717 + 2.54i)T \)
good5 \( 1 + (2.43 - 1.40i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.29 + 1.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.59 - 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.35 + 1.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.274 + 0.474i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.98 - 1.14i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.72 - 6.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.76T + 31T^{2} \)
37 \( 1 + (-1.81 + 3.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.12 + 4.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.93 - 2.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + (-2.53 - 4.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.35T + 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (-4.29 + 2.47i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.61iT - 79T^{2} \)
83 \( 1 + (-0.719 - 1.24i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.24 + 1.87i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.7 - 9.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.092414996627766692678740500308, −7.88618269864081693181052350526, −6.81605661693353364072840435400, −6.52106404358757875047907882274, −5.29399077609032675675092949780, −4.36415787470673696403240270810, −3.48956198232114320169468273963, −3.15733626392976714106599872550, −1.49331542928699595576372638428, −0.16360219633305262118377147027, 1.20290281675010813144628475604, 2.60144458788819238257945414801, 3.46149878872756973258079051069, 4.30738049549329231342236759914, 5.18223045671667269722474919957, 5.89835278834385212765437403292, 6.68354152299891734350294756932, 7.86542285891648540606239611085, 8.310078996469048314089474126994, 8.604868965088810315252754422769

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