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Properties

Label 2-3024-63.58-c1-0-7
Degree 22
Conductor 30243024
Sign −0.579−0.814i-0.579 - 0.814i
Analytic cond. 24.146724.1467
Root an. cond. 4.913934.91393
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 + 2.31i)5-s + (−0.581 − 2.58i)7-s + (−0.682 + 1.18i)11-s + (−2.75 + 4.77i)13-s + (1.23 + 2.14i)17-s + (2.19 − 3.80i)19-s + (2.34 + 4.06i)23-s + (−1.05 + 1.83i)25-s + (−2.94 − 5.10i)29-s − 3.11·31-s + (5.18 − 4.78i)35-s + (−3.15 + 5.46i)37-s + (−1.38 + 2.40i)41-s + (4.87 + 8.45i)43-s − 10.0·47-s + ⋯
L(s)  = 1  + (0.596 + 1.03i)5-s + (−0.219 − 0.975i)7-s + (−0.205 + 0.356i)11-s + (−0.764 + 1.32i)13-s + (0.300 + 0.520i)17-s + (0.503 − 0.872i)19-s + (0.488 + 0.846i)23-s + (−0.211 + 0.366i)25-s + (−0.547 − 0.948i)29-s − 0.559·31-s + (0.877 − 0.809i)35-s + (−0.518 + 0.898i)37-s + (−0.216 + 0.375i)41-s + (0.744 + 1.28i)43-s − 1.46·47-s + ⋯

Functional equation

Λ(s)=(3024s/2 ΓC(s) L(s)=((−0.579−0.814i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3024s/2 ΓC(s+1/2) L(s)=((−0.579−0.814i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30243024    =    24⋅33⋅72^{4} \cdot 3^{3} \cdot 7
Sign: −0.579−0.814i-0.579 - 0.814i
Analytic conductor: 24.146724.1467
Root analytic conductor: 4.913934.91393
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3024(2305,⋅)\chi_{3024} (2305, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3024, ( :1/2), −0.579−0.814i)(2,\ 3024,\ (\ :1/2),\ -0.579 - 0.814i)

Particular Values

L(1)L(1) ≈\approx 1.2675831821.267583182
L(12)L(\frac12) ≈\approx 1.2675831821.267583182
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
7 1+(0.581+2.58i)T 1 + (0.581 + 2.58i)T
good5 1+(−1.33−2.31i)T+(−2.5+4.33i)T2 1 + (-1.33 - 2.31i)T + (-2.5 + 4.33i)T^{2}
11 1+(0.682−1.18i)T+(−5.5−9.52i)T2 1 + (0.682 - 1.18i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.75−4.77i)T+(−6.5−11.2i)T2 1 + (2.75 - 4.77i)T + (-6.5 - 11.2i)T^{2}
17 1+(−1.23−2.14i)T+(−8.5+14.7i)T2 1 + (-1.23 - 2.14i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.19+3.80i)T+(−9.5−16.4i)T2 1 + (-2.19 + 3.80i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.34−4.06i)T+(−11.5+19.9i)T2 1 + (-2.34 - 4.06i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.94+5.10i)T+(−14.5+25.1i)T2 1 + (2.94 + 5.10i)T + (-14.5 + 25.1i)T^{2}
31 1+3.11T+31T2 1 + 3.11T + 31T^{2}
37 1+(3.15−5.46i)T+(−18.5−32.0i)T2 1 + (3.15 - 5.46i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.38−2.40i)T+(−20.5−35.5i)T2 1 + (1.38 - 2.40i)T + (-20.5 - 35.5i)T^{2}
43 1+(−4.87−8.45i)T+(−21.5+37.2i)T2 1 + (-4.87 - 8.45i)T + (-21.5 + 37.2i)T^{2}
47 1+10.0T+47T2 1 + 10.0T + 47T^{2}
53 1+(−1.47−2.56i)T+(−26.5+45.8i)T2 1 + (-1.47 - 2.56i)T + (-26.5 + 45.8i)T^{2}
59 1−3.55T+59T2 1 - 3.55T + 59T^{2}
61 1−1.32T+61T2 1 - 1.32T + 61T^{2}
67 1+8.29T+67T2 1 + 8.29T + 67T^{2}
71 1−12.3T+71T2 1 - 12.3T + 71T^{2}
73 1+(1.11+1.93i)T+(−36.5+63.2i)T2 1 + (1.11 + 1.93i)T + (-36.5 + 63.2i)T^{2}
79 1+12.8T+79T2 1 + 12.8T + 79T^{2}
83 1+(−5.15−8.93i)T+(−41.5+71.8i)T2 1 + (-5.15 - 8.93i)T + (-41.5 + 71.8i)T^{2}
89 1+(7.73−13.3i)T+(−44.5−77.0i)T2 1 + (7.73 - 13.3i)T + (-44.5 - 77.0i)T^{2}
97 1+(2.55+4.42i)T+(−48.5+84.0i)T2 1 + (2.55 + 4.42i)T + (-48.5 + 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.314542869924263319188302716049, −8.041429785512737603567459993368, −7.25136511944887082538940634988, −6.85292068613647829773260370682, −6.16906993751779257103955456633, −5.08496389946251228699718247530, −4.28427426124649796407915952941, −3.33887562874485420143445751072, −2.46987844001277422286297414051, −1.44471555125280190272201085556, 0.38160215509133737901614256665, 1.68603578340428136214318679872, 2.71584519697591607003224589904, 3.53934170048792393072348070362, 4.95968025564493667740934831280, 5.41017512758756767933005028238, 5.80639870343907328399972722449, 7.01800266878837032331893128018, 7.81973056923527748117577056319, 8.663617493232305867412811999287

Graph of the ZZ-function along the critical line