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Properties

Label 2-1512-63.25-c1-0-0
Degree 22
Conductor 15121512
Sign −0.709−0.705i-0.709 - 0.705i
Analytic cond. 12.073312.0733
Root an. cond. 3.474673.47467
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  + (−0.918 + 1.59i)5-s + (−0.361 − 2.62i)7-s + (−1.54 − 2.68i)11-s + (2.40 + 4.16i)13-s + (−1.87 + 3.24i)17-s + (−2.71 − 4.70i)19-s + (−3.97 + 6.89i)23-s + (0.813 + 1.40i)25-s + (0.325 − 0.563i)29-s + 1.03·31-s + (4.50 + 1.83i)35-s + (0.873 + 1.51i)37-s + (−2.52 − 4.36i)41-s + (−6.09 + 10.5i)43-s + 4.61·47-s + ⋯
L(s)  = 1  + (−0.410 + 0.711i)5-s + (−0.136 − 0.990i)7-s + (−0.466 − 0.808i)11-s + (0.666 + 1.15i)13-s + (−0.453 + 0.786i)17-s + (−0.622 − 1.07i)19-s + (−0.829 + 1.43i)23-s + (0.162 + 0.281i)25-s + (0.0604 − 0.104i)29-s + 0.186·31-s + (0.760 + 0.309i)35-s + (0.143 + 0.248i)37-s + (−0.393 − 0.682i)41-s + (−0.929 + 1.61i)43-s + 0.672·47-s + ⋯

Functional equation

Λ(s)=(1512s/2 ΓC(s) L(s)=((−0.709−0.705i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1512s/2 ΓC(s+1/2) L(s)=((−0.709−0.705i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15121512    =    23⋅33⋅72^{3} \cdot 3^{3} \cdot 7
Sign: −0.709−0.705i-0.709 - 0.705i
Analytic conductor: 12.073312.0733
Root analytic conductor: 3.474673.47467
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1512(1369,⋅)\chi_{1512} (1369, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1512, ( :1/2), −0.709−0.705i)(2,\ 1512,\ (\ :1/2),\ -0.709 - 0.705i)

Particular Values

L(1)L(1) ≈\approx 0.61231369500.6123136950
L(12)L(\frac12) ≈\approx 0.61231369500.6123136950
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+(0.361+2.62i)T 1 + (0.361 + 2.62i)T
good5 1+(0.918−1.59i)T+(−2.5−4.33i)T2 1 + (0.918 - 1.59i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.54+2.68i)T+(−5.5+9.52i)T2 1 + (1.54 + 2.68i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2.40−4.16i)T+(−6.5+11.2i)T2 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2}
17 1+(1.87−3.24i)T+(−8.5−14.7i)T2 1 + (1.87 - 3.24i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.71+4.70i)T+(−9.5+16.4i)T2 1 + (2.71 + 4.70i)T + (-9.5 + 16.4i)T^{2}
23 1+(3.97−6.89i)T+(−11.5−19.9i)T2 1 + (3.97 - 6.89i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.325+0.563i)T+(−14.5−25.1i)T2 1 + (-0.325 + 0.563i)T + (-14.5 - 25.1i)T^{2}
31 1−1.03T+31T2 1 - 1.03T + 31T^{2}
37 1+(−0.873−1.51i)T+(−18.5+32.0i)T2 1 + (-0.873 - 1.51i)T + (-18.5 + 32.0i)T^{2}
41 1+(2.52+4.36i)T+(−20.5+35.5i)T2 1 + (2.52 + 4.36i)T + (-20.5 + 35.5i)T^{2}
43 1+(6.09−10.5i)T+(−21.5−37.2i)T2 1 + (6.09 - 10.5i)T + (-21.5 - 37.2i)T^{2}
47 1−4.61T+47T2 1 - 4.61T + 47T^{2}
53 1+(4.55−7.88i)T+(−26.5−45.8i)T2 1 + (4.55 - 7.88i)T + (-26.5 - 45.8i)T^{2}
59 1−5.79T+59T2 1 - 5.79T + 59T^{2}
61 1+4.81T+61T2 1 + 4.81T + 61T^{2}
67 1+14.4T+67T2 1 + 14.4T + 67T^{2}
71 1−5.00T+71T2 1 - 5.00T + 71T^{2}
73 1+(1.81−3.14i)T+(−36.5−63.2i)T2 1 + (1.81 - 3.14i)T + (-36.5 - 63.2i)T^{2}
79 1+14.3T+79T2 1 + 14.3T + 79T^{2}
83 1+(3.83−6.63i)T+(−41.5−71.8i)T2 1 + (3.83 - 6.63i)T + (-41.5 - 71.8i)T^{2}
89 1+(−5.76−9.99i)T+(−44.5+77.0i)T2 1 + (-5.76 - 9.99i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.04−1.80i)T+(−48.5−84.0i)T2 1 + (1.04 - 1.80i)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

−9.836984246029933207150585051536, −8.924782814413200175122733453721, −8.128218912206622742714048585531, −7.28579559672557891885959145038, −6.62411014058797412032210064569, −5.88701538634692788032419764732, −4.51434473295870697293694536632, −3.82164497379051396035567019265, −2.95438573127422286240971317541, −1.50152736652017719441902811481, 0.23734994505371332974361434345, 1.94744561100907289715647000971, 2.96536209258642349383044857445, 4.19708674610312290997571802440, 5.00010458303719611503768799998, 5.81492883805851386180600596126, 6.65424608869631129893589907273, 7.84278920998385155855116890952, 8.409589053741410420573621171354, 8.945099729721354150627616262592

Graph of the ZZ-function along the critical line